PRECIPITACIÓN EN PUEBLA

DESARROLLO

Tabla de Contenido

  • CONFIGURACIÓN DE LA NOTETBOOK
  • CARGAR DATOS
  • PARTIR LA SERIE PARA TESTEARLA
  • SERIE SIN TRANSFORMACIONES
    • ESTACIONARIEDAD
    • AUTOCORRELACIONES
    • DIFERENCIA ESTACIONAL
      • PRIMERA DIFERENCIA ESTACIONAL
        • ESTACIONARIEDAD
        • FAC Y FACP
      • SEGUNDA DIFERENCIA ESTACIONAL
        • FAC Y FACP
      • TERCERA DIFERENCIA ESTACIONAL
        • FAC Y FACP
      • COMPARACIÓN DE VARIANZAS CON DISTINTAS DIFERENCIAS ESTACIONALES
  • SERIE CON TRANSFORMACIÓN
    • ESTIMADOR PUNTUAL
    • INTERVALO DE CONFIANZA
    • SERIE TRANSFORMADA
    • ESTACIONARIEDAD
    • FAC Y FACP
      • DIFERENCIAS ESTACIONALES
        • PRIMERA DIFERENCIA ESTACIONAL
          • ESTACIONARIEDAD
          • FAC Y FACP
        • SEGUNDA DIFERENCIA ESTACIONAL
          • FAC Y FACP
        • TERCERA DIFERENCIA ESTACIONAL
          • FAC Y FACP
        • COMPARACIÓN DE VARIANZAS CON DISTINTAS DIFERENCIAS ESTACIONALES
  • BUSQUEDA DE MODELOS
    • PMDARIMA
    • RESULTADOS de 03-01-Encontrar-Modelo-pq
    • RESULTADOS de 03-02-Encontrar-Modelo-Pq
    • RESULTADOS de 03-03-Encontrar-Modelo-pQ
    • RESULTADOS de 03-04-Encontrar-Modelo-PQ
    • MODELOS ORDENADOS POR SU $\text{AIC}$:
  • MODELOS PROPUESTOS
    • MODELOS PARA LA SERIE SIN TRANSFORMAR
      • PRIMER MODELO
        • VERIFICACIÓN DE SUPUESTOS
          • PRINCIPIO DE PARSIMONIA
          • MODELO ADMISIBLE
          • RESIDUOS INDEPENDIENTES
          • RESIDUOS CON MEDIA CERO
          • RESIDUOS CON VARIANZA CONSTANTE
          • RESIDUOS CON DISTRIBUCIÓN NORMAL
          • GRÁFICO DE RESIDUOS
      • SEGUNDO MODELO
        • VERIFICACIÓN DE SUPUESTOS
          • PRINCIPIO DE PARSIMONIA
          • MODELO ADMISIBLE
          • RESIDUOS INDEPENDIENTES
          • RESIDUOS CON MEDIA CERO
          • RESIDUOS CON VARIANZA CONSTANTE
          • RESIDUOS CON DISTRIBUCIÓN NORMAL
          • GRÁFICO DE RESIDUOS
      • TERCER MODELO
        • VERIFICACIÓN DE SUPUESTOS
          • PRINCIPIO DE PARSIMONIA
          • MODELO ADMISIBLE
          • RESIDUOS INDEPENDIENTES
          • RESIDUOS CON MEDIA CERO
          • RESIDUOS CON VARIANZA CONSTANTE
          • RESIDUOS CON DISTRIBUCIÓN NORMAL
          • GRÁFICO DE RESIDUOS
      • OTROS MODELOS QUE SE CREYÓ QUE PODÍAN SER
    • MODELO PARA LA SERIE TRANSFORMADA
      • PRIMER MODELO
        • VERIFICACIÓN DE SUPUESTOS
          • RESIDUOS INDEPENDIENTES
          • RESIDUOS CON MEDIA CERO
          • GRÁFICO DE RESIDUOS
      • SEGUNDO MODELO
        • VERIFICACIÓN DE SUPUESTOS
          • RESIDUOS INDEPENDIENTES
      • TERCER MODELO
        • VERIFICACIÓN DE SUPUESTOS
          • PRINCIPIO DE PARSIMONIA
          • MODELO ADMISIBLE
          • RESIDUOS INDEPENDIENTES
          • RESIDUOS CON MEDIA CERO
          • RESIDUOS CON VARIANZA CONSTANTE
          • RESIDUOS CON DISTRIBUCIÓN NORMAL
          • GRÁFICO DE RESIDUOS
  • MODELOS SELECCIONADOS
    • MODELO SELECCIONADO OBTENIDO CON LA SERIE SIN TRANSFORMAR
    • MODELO SELECCIONADO OBTENIDO CON LA SERIE TRANSFORMADA
  • PRÓNOSTICO
    • MÉTODO FORECAST
      • FORECAST
        • INVERSA DE LA TRASNFORMACIÓN PARA EL PRONÓSTICO
      • INTERVALOS DE CONFIANZA
        • INVERSA DE LA TRANSFORMACIÓN PARA EL IC
          • RESULTADO OBTENIDO CON SERIE SIN TRANSFORMAR
          • RESULTADO CON LA SERIE TRANSFORMADA
          • COMPARACIÓN
    • PRONÓSTICO ÓPTIMO
      • OBTENCIÓN DEL PRONÓSTICO ÓPTIMO
        • DESARROLLO PARA EL PRONÓSTICO ÓPTIMO
        • PRONÓSTICO ÓPTIMO
      • INTERVALO DE CONFIANZA
        • CALCULO DE LAS $\psi$
        • DESVIACIÓN ESTANDAR DEL RUIDO BLANCO
        • INTERVALOS DE PREDICCIÓN
    • COMPARACIÓN
  • ACTUALIZACIÓN DE PRONÓSTICO
    • CON PYTHON
    • MANUAL
    • COMPARACIÓN

CONFIGURACIÓN DE LA NOTETBOOK¶

CARGAR DATOS¶

Precipitación v4:
https://www.ncei.noaa.gov/products/land-based-station/global-historical-climatology-network-monthly https://www.ncei.noaa.gov/data/ghcnm/v4/precipitation/ https://www.ncei.noaa.gov/pub/data/ghcn/v4/products/StationPlots/MX/

Nombres:
https://www.ncei.noaa.gov/pub/data/ghcn/v4/
https://www.ncei.noaa.gov/data/ghcnm/v4/precipitation/doc/

Ubicación de la estación: 19.0000, -98.1833

No description has been provided for this image
No description has been provided for this image

PARTIR LA SERIE PARA TESTEARLA¶

SERIE SIN TRANSFORMACIONES¶

Como la vamos a comparar con la transformacion de Yeo-Johnson, estandarizamos para comparar los modelos.

ESTACIONARIEDAD¶

(-6.150368236022414,
 7.588129761677003e-08,
 11,
 648,
 {'1%': -3.4404817800778034,
  '5%': -2.866010569916275,
  '10%': -2.569150763698369},
 1333.5800521506967)
Estadístico ADF         = -6.150  
Valor-p                 = 7.59e-08  
Número de rezagos       = 11  
Número de observaciones = 648  
Valores críticos:
    1%  -> -3.4405
    5%  -> -2.8660  
    10% -> -2.5692
Log-likelihood          = 9999.935
  • El estadístico ADF es $-6.150$, menor que los valores críticos a los niveles del 1%.
  • El valor-p es $7.59 × 10^{-8}$ mucho menor al nivel de significancia $\alpha$ = 0.05.

Por lo tanto, se rechaza la hipótesis nula, es decir, se concluye que la serie es estacionaria.

AUTOCORRELACIONES¶

No description has been provided for this image
Valores de autocorrelacion significativos:
r1: 0.5114347926277153
r2: 0.23788150833700863
r4: -0.31136216826642676
r5: -0.5289967604156934
r6: -0.5765988530134252
r7: -0.4540298620448335
r8: -0.24420551667402526
r10: 0.2960004737880804
r11: 0.5240658344251691
r12: 0.6173499572874946
r13: 0.424401244621075
r14: 0.18265000690415573
r16: -0.31903920298321264
r17: -0.514286815393446
r18: -0.527829591696689
r19: -0.4187324328138189
r22: 0.30067304144252044
r23: 0.5028304240974979
r24: 0.5626631891391944
r25: 0.3994834359095428
r28: -0.34516091598549786
r29: -0.48547226384609826
r30: -0.5033566062877032
r31: -0.38761684940293784
r34: 0.3085645527226526
r35: 0.4965812730150526
r36: 0.553098035132298
r37: 0.36241966295184613
r40: -0.3454697704347722
r41: -0.4674618422218585
r42: -0.46761339396960605
r43: -0.34243418620086286
r46: 0.3414843279972229
r47: 0.48860016886221763
r48: 0.5000880408130988
Valores de autocorrelacion parcial significativos:
rho 1: 0.5122108697030228
rho 3: -0.1371024981440388
rho 4: -0.35568794886746186
rho 5: -0.3390668723583037
rho 6: -0.26990992305506756
rho 7: -0.13651941979413404
rho 8: -0.082068919602615
rho 11: 0.19446853660047775
rho 12: 0.25021094781029624
rho 19: -0.0924235210146644
rho 24: 0.12444505947195694
rho 32: -0.08477112361318034
rho 36: 0.11301403546356209

Aunque la serie es estacionaria, dado que las autocorrelaciones simples decrecen muy muy lento, hacemos una primera diferencia estacional.

DIFERENCIA ESTACIONAL¶

La diferencia estacional de orden $D$, con una periodicidad estacional $s$, se define de forma recursiva aplicando el operador $\nabla_s$ múltiples veces:

$$ \nabla_s^D X_t = (1 - B^s)^D X_t $$

donde:

  • $D$ es el orden de la diferencia estacional,
  • $B$ es el operador de rezago tal que $B^k X_t = y_{X-k}$,
  • $\nabla_s^D$ representa aplicar $D$ veces la diferencia estacional.

Para esta serie, el orden de la diferencia estacional es $D = 1$ y la periodicidad estacional es $s = 12$:

$$ \nabla_{12}^1 \text{T} (X_t) = \text{T} (X_t) - \text{T} (X_{t-12}) $$

Sea entonces $Wt = \nabla_{12}^1 \text{T} (X_t) = \text{T} (X_t) - \text{T} (X_{t-12})$

PRIMERA DIFERENCIA ESTACIONAL¶

No description has been provided for this image

ESTACIONARIEDAD¶

(-10.871408688779647,
 1.3655020494523785e-19,
 11,
 636,
 {'1%': -3.4406737255613256,
  '5%': -2.866095119842903,
  '10%': -2.5691958123689727},
 1448.5800337609196)
Estadístico ADF         = -10.871  
Valor-p                 = 1.37e-19  
Número de rezagos       = 11  
Número de observaciones = 636  
Valores críticos:
    1%  -> -3.4407
    5%  -> -2.8661
    10% -> -2.5692
Log-likelihood          = 9952.186
  • El estadístico ADF es $-10.871$, menor que los valores críticos a los niveles del 1%.
  • El valor-p es $1.37 × 10^{-19}$ mucho menor al nivel de significancia $\alpha$ = 0.05.

Por lo tanto, se rechaza la hipótesis nula, es decir, se concluye que la serie es estacionaria.

FAC Y FACP¶

No description has been provided for this image

Se sigue notando la estacionalidad de periodicidad 12.

Valores de autocorrelacion significativos:
r1: 0.0849542779953831
r5: -0.09455404757011154
r12: -0.414183329502961
r69: 0.13811523521698316
r73: -0.1125610007638793
r82: -0.09936326169869979
Valores de autocorrelacion parcial significativos:
rho 1: 0.08508558290727707
rho 5: -0.08389897753926695
rho 12: -0.435832204242704
rho 17: -0.12023508334631498
rho 24: -0.30097443805877616
rho 29: -0.10198920198664953
rho 36: -0.15950168604080855
rho 48: -0.14036369491687178
rho 57: -0.0916871776051421
rho 60: -0.12396066058923692
rho 61: 0.13800141862943816
rho 69: 0.10269095820413293
rho 72: -0.09462958643919762
rho 82: -0.08002594037040592
rho 85: -0.0819832551248085
rho 93: 0.0922037887338383
rho 109: 0.09582379018763905
rho 113: -0.09105599732819779
rho 119: 0.0917177196667104
rho 120: -0.1355872480803115

SEGUNDA DIFERENCIA ESTACIONAL¶

FAC Y FACP¶

No description has been provided for this image
Valores de autocorrelacion significativos:
r12: -0.6150182129084292
r57: -0.10922275236044279
r69: 0.16716305731426437
r73: -0.11365881167509073
r81: -0.12386243150076603
r82: -0.11601226324982332
Valores de autocorrelacion parcial significativos:
rho 12: -0.6294838987246432
rho 17: -0.09052048792417981
rho 24: -0.5103011546739746
rho 29: -0.093217025773514
rho 36: -0.3551238626796082
rho 48: -0.2751156800940471
rho 57: -0.108916168798713
rho 60: -0.2379289221214959
rho 61: 0.20095345338550902
rho 65: -0.0834129475389058
rho 72: -0.29986531752257994
rho 73: 0.302496719075776
rho 74: -0.15214692323832477
rho 75: 0.12158013142041356
rho 77: -0.21501383907290988
rho 79: -0.08587576253968264
rho 80: 0.10575443412592936
rho 81: -0.11295475178675196
rho 83: 0.1412351900166441
rho 84: -0.47389037358187136
rho 85: 0.8052856123151159
rho 86: -2.7039482046544805
rho 87: -1.3736483987654273
rho 88: 1.0019992843716
rho 89: 235.89044017004235
rho 90: -0.9971937437125793
rho 91: 0.48021486078669995
rho 93: -0.20196306057247534
rho 94: 0.28532937400900626
rho 95: -0.23402172704695934
rho 101: 0.25176068174227584
rho 102: -0.5167154340590121
rho 103: 0.7690418805288879
rho 104: -1.3401560972346027
rho 105: -2.166656395549073
rho 106: 1.1988394283045958
rho 107: 2.0506797927028866
rho 108: -1.2599924429780232
rho 109: -2.603184507263249
rho 110: 0.5135974713732451
rho 111: 0.2590435314099863
rho 113: -0.3628723780659208
rho 114: 0.4207095680305757
rho 115: -0.3885886584070735
rho 116: 0.2697703099821219
rho 119: -0.12676720968125227
rho 120: 0.16418782499731185

TERCERA DIFERENCIA ESTACIONAL¶

FAC Y FACP¶

No description has been provided for this image
Valores de autocorrelacion significativos:
r12: -0.6989860750076662
r24: 0.19148304290921273
r57: -0.13077245780878802
r69: 0.18180783859402475
r81: -0.1446388376312966
Valores de autocorrelacion parcial significativos:
rho 12: -0.7131085439918823
rho 24: -0.6165329897683706
rho 29: -0.08067623128823141
rho 36: -0.5171728973407944
rho 37: 0.1982055108755419
rho 40: 0.10015883076893528
rho 41: -0.09228473683010521
rho 47: 0.09394099253473005
rho 48: -0.4929991451837127
rho 49: 0.27281984306019774
rho 50: -0.09255185942814254
rho 51: -0.10940745619258534
rho 52: 0.34514660641128075
rho 53: -0.3969949656195508
rho 54: 0.33091157283957406
rho 55: -0.24725105044265083
rho 57: -0.1678397209757971
rho 58: -0.41066728720157253
rho 59: 1.4752062518993367
rho 60: 3.447013197287627
rho 61: -0.965945384029374
rho 62: 6.762979431106962
rho 63: 1.0827131685786548
rho 64: -0.8201163848967379
rho 65: 1.5200084948857941
rho 66: 1.3863162790341865
rho 67: -2.332119329595056
rho 68: -1.2962242829344859
rho 69: 0.5251991629964926
rho 70: 0.12535213223464997
rho 71: -0.43609447528833045
rho 72: 0.35806234464280917
rho 73: -0.1138999259876181
rho 74: -0.10235078255767406
rho 75: 0.31494875075818757
rho 76: -0.20483724449372892
rho 77: 0.0921074399984271
rho 78: -0.08928014769113522
rho 79: 0.0854845471954933
rho 80: -0.28927169831303046
rho 81: 0.20932339081585724
rho 82: -0.11854528891185322
rho 84: 0.19492618448545254
rho 87: 0.21562177268699964
rho 88: -0.19739536579003492
rho 89: 0.1375019109969488
rho 90: -0.21573215769368453
rho 91: 0.13649794688884304
rho 92: -0.4130856229291036
rho 93: 0.5769402061332968
rho 94: -1.1748169850119188
rho 95: -6.0784018693922
rho 96: 0.9057557611685417
rho 97: -0.5526621327578615
rho 98: 0.3652101944200118
rho 99: -0.14934082325742695
rho 100: -0.20792999327433961
rho 101: 0.32248566533747575
rho 102: -0.43340585213601085
rho 103: 0.35823263199853145
rho 104: -0.16127048470991623
rho 105: -0.09034092543015515
rho 106: 0.8633154494572077
rho 107: -9.435548190300198
rho 108: -1.0889219702203907
rho 109: 1.4801001973243877
rho 110: 2.1038789559878754
rho 111: -0.7905244792702399
rho 112: 0.6723557185188808
rho 113: -0.9378271953920853
rho 114: 5.770582312925257
rho 115: 0.947663800814546
rho 116: 3.6394621713184363
rho 117: -0.8146958649632807
rho 118: 0.18972318689108425
rho 119: 0.4988023348063774
rho 120: -0.48455031039030716

COMPARACIÓN DE VARIANZAS CON DISTINTAS DIFERENCIAS ESTACIONALES¶

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Nos quedamos con la primera diferencia estacional, porque tiene una mejor forma su FAC y FACP, además de tener menor varianza.

SERIE CON TRANSFORMACIÓN¶

Se utilizó la transformación Yeo-Johnson para aproximar la normalidad de los datos, lo que permite que los modelos ajustados sean más robustos y que las inferencias estadísticas sean confiables, además de ayudar a estabilizar la varianza. Esta transformación se eligió porque existen meses con precipitación igual a cero, un caso donde la transformación Box-Cox no es aplicable. Yeo-Johnson acepta valores cero sin necesidad de modificar los datos con constantes arbitrarias, preservando así su integridad y significado original.

La precipitación es una variable altamente sesgada, con muchos meses de poca o ninguna lluvia y pocos meses con lluvias abundantes. Esto genera una distribución con cola larga y asimétrica que dificulta el modelado estadístico y la interpretación.

La transformación Yeo-Johnson corrige esta asimetría y acerca la distribución a una normal, mejorando los supuestos de los modelos estadísticos y aumentando la precisión en predicciones y simulaciones.

Además, la precipitación presenta varianza heterogénea, con meses en que la lluvia varía mucho respecto a otros. Estabilizar esta varianza es fundamental para aplicar modelos estadísticos y de series de tiempo, ya que evita que la heterocedasticidad afecte la validez de las inferencias. La transformación Yeo-Johnson contribuye a esta estabilización de la varianza.

Definición matemática de la transformación Yeo-Johnson, como se encuentra en la literatura estadística (propuesta por Ingram Olkin y I. Paul Yeo & R. J. Johnson en 2000).

Sea $x \in \mathbb{R}$ un valor (puede ser negativo, cero o positivo). La transformación $T(x; \lambda)$ se define como:

$$ T(x; \lambda) = \begin{cases} \frac{[(x + 1)^\lambda - 1]}{\lambda} & \text{si } x \geq 0, \, \lambda \ne 0 \\ \log(x + 1) & \text{si } x \geq 0, \, \lambda = 0 \\ -\frac{[(-x + 1)^{2 - \lambda} - 1]}{2 - \lambda} & \text{si } x < 0, \, \lambda \ne 2 \\ -\log(-x + 1) & \text{si } x < 0, \, \lambda = 2 \end{cases} $$

ESTIMADOR PUNTUAL¶

Lambda estimado: [0.23849441]

INTERVALO DE CONFIANZA¶

IC 95% para λ: (0.2082, 0.2708)
No description has been provided for this image

SERIE TRANSFORMADA¶

No description has been provided for this image

ESTACIONARIEDAD¶

(-6.885194656270816,
 1.4004074124320326e-09,
 16,
 643,
 {'1%': -3.440560883168159,
  '5%': -2.8660454146233434,
  '10%': -2.569169329058723},
 1139.5328800342036)
Estadístico ADF         = -6.885  
Valor-p                 = 1.40e-09  
Número de rezagos       = 16  
Número de observaciones = 643  
Valores críticos:
    1%  -> -3.4406
    5%  -> -2.8660
    10% -> -2.5692
Log-likelihood          = 3742.248
  • El estadístico ADF es $-6.885$, menor que los valores críticos a los niveles del 1%.
  • El valor-p es $1.40 × 10^{-9}$ mucho menor al nivel de significancia $\alpha$ = 0.05.

Por lo tanto, se rechaza la hipótesis nula, es decir, se concluye que la serie es estacionaria.

FAC Y FACP¶

No description has been provided for this image
Valores de autocorrelacion significativos:
r1: 0.6331507478065659
r2: 0.3325589848089995
r4: -0.3429619382624403
r5: -0.6128788441981868
r6: -0.7155687723661789
r7: -0.5667143985229659
r8: -0.28172958031763784
r10: 0.34639100385471927
r11: 0.6139734232526982
r12: 0.6833003454277807
r13: 0.5238197525884694
r14: 0.23526370655336934
r16: -0.37665249496039976
r17: -0.6133232329123165
r18: -0.6522050881633413
r19: -0.49864844118107393
r22: 0.3581357207867556
r23: 0.6051041828734448
r24: 0.632952748472827
r25: 0.4787907060076014
r28: -0.3830626619493817
r29: -0.5771240646474324
r30: -0.6219569734501011
r31: -0.44116167635231773
r34: 0.3905997714697265
r35: 0.5842211455257492
r36: 0.606315712870404
r37: 0.41306326727063364
r40: -0.39053448733579516
r41: -0.5677702589671872
r42: -0.5689341833781766
r43: -0.3981104753267791
r46: 0.3759283566696746
r47: 0.5704749093905213
r48: 0.5490665087242427
Valores de autocorrelacion parcial significativos:
rho 1: 0.6341115228411734
rho 2: -0.11461335623878138
rho 3: -0.3000454427793004
rho 4: -0.33135817233932896
rho 5: -0.38109982701434264
rho 6: -0.32479269111279224
rho 7: -0.08061163300684875
rho 11: 0.2000309079008965
rho 12: 0.1532581642408276
rho 17: -0.12292322538615731
rho 22: -0.08358889406877636
rho 23: 0.12630684698989092
rho 30: -0.1028445923709159
rho 47: 0.09892532485023141

DIFERENCIAS ESTACIONALES¶

Por la FAC y la FACP, la serie transformada también necesita diferencias estacionales.

PRIMERA DIFERENCIA ESTACIONAL¶

No description has been provided for this image
ESTACIONARIEDAD¶
(-9.030402747697376,
 5.464639337632109e-15,
 16,
 631,
 {'1%': -3.440755866431696,
  '5%': -2.86613130039063,
  '10%': -2.569215089800357},
 1308.464428619442)
Estadístico ADF         = -10.871  
Valor-p                 = 1.37e-19  
Número de rezagos       = 11  
Número de observaciones = 636  
Valores críticos:
    1%  -> -3.4407
    5%  -> -2.8661
    10% -> -2.5692
Log-likelihood          = 9952.186
  • El estadístico ADF es $-10.871$, menor que los valores críticos a los niveles del 1%.
  • El valor-p es $1.37 × 10^{-19}$ mucho menor al nivel de significancia $\alpha$ = 0.05.

Por lo tanto, se rechaza la hipótesis nula, es decir, se concluye que la serie es estacionaria.

FAC Y FACP¶
No description has been provided for this image
Valores de autocorrelacion significativos:
r1: 0.1881206416574384
r2: 0.12601374062799692
r6: -0.10023577930193152
r12: -0.40245032162240263
r33: 0.09802046981427195
r45: -0.11985655705785987
Valores de autocorrelacion parcial significativos:
rho 1: 0.18841139999075743
rho 2: 0.09425081653205467
rho 6: -0.0848567418000625
rho 12: -0.4319892618507997
rho 13: 0.08136260875048439
rho 17: -0.11385209887045368
rho 22: -0.08401445717662205
rho 24: -0.2809725408252081
rho 25: 0.10755279636113212
rho 30: -0.09673228306484968
rho 33: 0.1105360450418002
rho 36: -0.14602967562386657
rho 42: -0.11769525648448687
rho 46: -0.09378255882994088
rho 48: -0.16710968828081296
rho 50: 0.09151867173223789
rho 55: 0.09278816335947565
rho 60: -0.09971581325023068
rho 72: -0.13002571917786468
rho 78: -0.07964448430687596
rho 84: -0.07986957194068202
rho 92: -0.13966165907823416
rho 93: 0.11600081821072342
rho 94: -0.1221857229746919
rho 113: -0.08611614522897519
rho 118: -0.08199830520693245
rho 120: -0.11627799482364788

SEGUNDA DIFERENCIA ESTACIONAL¶

FAC Y FACP¶
No description has been provided for this image
Valores de autocorrelacion significativos:
r1: 0.1509169821138114
r2: 0.09453473815881087
r12: -0.618494697570679
r13: -0.11450254134076049
r33: 0.12874644766913002
r45: -0.13676859805248878
r46: -0.11237170141253933
Valores de autocorrelacion parcial significativos:
rho 1: 0.15115464665257325
rho 12: -0.63026665230029
rho 14: 0.10193023549147724
rho 22: -0.10303705027323279
rho 24: -0.49273561388444265
rho 25: 0.12655475900658333
rho 36: -0.3174803537947576
rho 37: 0.12278759137098168
rho 42: -0.11657573095130833
rho 46: -0.11977143455916281
rho 48: -0.269726654008099
rho 50: 0.08236706657389534
rho 54: -0.12105303180693465
rho 58: -0.09682438702973352
rho 60: -0.17197052470804497
rho 61: 0.1271082992001105
rho 64: -0.08425018878431509
rho 66: -0.1334769248947342
rho 67: 0.10772337653637902
rho 70: -0.18205599974434547
rho 71: 0.15370692142158723
rho 72: -0.28367137696962097
rho 73: 0.28767316323135766
rho 74: -0.264867047788194
rho 75: 0.2758140373276588
rho 76: -0.49750333625335225
rho 77: 0.9457796838602422
rho 78: -19.481457329466302
rho 79: -1.0525724850043823
rho 80: 0.6171731402566649
rho 81: -0.3683891573808938
rho 82: 0.19768676162895707
rho 83: -0.10963040531813105
rho 85: -0.1441567606193653
rho 86: 0.2537929133818078
rho 87: -0.19325991698191852
rho 89: -0.08086510996156981
rho 92: 0.12516612507611838
rho 93: -0.10383117667192988
rho 95: -0.08858698291488463
rho 96: -0.08310750719586882
rho 98: 0.20514831945998865
rho 99: -0.19400587794851393
rho 102: -0.12948019476807293
rho 104: 0.19099339964326337
rho 105: -0.292576247094976
rho 108: -0.18822628009595083
rho 110: 0.30869856539482576
rho 111: -0.5129270744468297
rho 112: 0.09159365800962423
rho 113: 0.3763410499243059
rho 114: -0.7951142842424803
rho 115: 1.169936457430537
rho 116: -5.318688058804201
rho 117: -2.443886325406397
rho 118: 1.409634595547863
rho 119: -0.8889029696410261
rho 120: 0.45057413786733763

TERCERA DIFERENCIA ESTACIONAL¶

FAC Y FACP¶
No description has been provided for this image
Valores de autocorrelacion significativos:
r1: 0.14142525988694732
r11: -0.08618353748437811
r12: -0.7006965212651104
r13: -0.12299455598817151
r24: 0.21088909322401359
r33: 0.14799260300636627
r34: 0.124164682223292
r45: -0.13840321103959075
Valores de autocorrelacion parcial significativos:
rho 1: 0.14165226672464704
rho 11: -0.08572854737384034
rho 12: -0.7145741931403762
rho 14: 0.10244937563798918
rho 21: -0.0974412193826169
rho 22: -0.11958849854177123
rho 24: -0.590999784962588
rho 25: 0.1155406109341616
rho 36: -0.42596863520035666
rho 37: 0.18734873402939595
rho 42: -0.1733829621796275
rho 46: -0.1132553928924406
rho 48: -0.39708277000220765
rho 49: 0.17397913808655488
rho 54: -0.3269257232330364
rho 55: 0.21559931119807146
rho 56: -0.19500988553521115
rho 58: -0.22348004365902227
rho 59: 0.25219826387781036
rho 60: -0.7375993663060407
rho 61: 2.2537091363236095
rho 62: 1.650176227081085
rho 63: -0.6968349029252933
rho 64: 0.3967431321320999
rho 65: -0.2123951680489383
rho 66: -0.20497281303308482
rho 68: 0.23621005007258603
rho 69: -0.8189484098245967
rho 70: 1.3011193720883465
rho 71: 1.4316135793533165
rho 72: -5.898535785106428
rho 73: -1.3199340565173563
rho 74: -0.6772780595533806
rho 75: 0.5245892189329673
rho 76: -0.2815596779482279
rho 77: -0.4521464433566569
rho 78: 0.30058897281376856
rho 79: -0.4164535343317919
rho 80: -0.11859502105379839
rho 81: -0.15366212304252277
rho 82: -0.2922622187005957
rho 83: -0.4928118125453261
rho 84: -0.14977995949488318
rho 85: -0.8927516420039372
rho 86: -5.928900128261838
rho 87: 1.2117900757453872
rho 88: 0.5145451003205691
rho 89: 0.24638896698250676
rho 90: 0.5283271371105461
rho 92: 0.3314944963217833
rho 93: 0.11506584057595354
rho 94: 0.21220949431027086
rho 95: -0.12891904223431166
rho 96: 0.5141635439947064
rho 97: -0.6188892690196736
rho 98: 1.7101159305305729
rho 99: 2.12107036942252
rho 100: -0.6062698236020745
rho 101: 0.46103150897488143
rho 102: -0.1569944090737856
rho 103: 0.2535125462534724
rho 104: -0.21007258043287413
rho 105: 0.3130566565661735
rho 107: 0.1210008208606222
rho 109: 0.1042542157962568
rho 110: -0.17221115272098989
rho 111: 0.2646586117130243
rho 112: -0.11554768465356624
rho 114: 0.18496589221568
rho 115: 0.10819324030267996
rho 116: -0.15857098929802355
rho 117: 0.3781749655329778
rho 118: -0.1243216624045143
rho 120: 0.20596573821321837

COMPARACIÓN DE VARIANZAS CON DISTINTAS DIFERENCIAS ESTACIONALES¶

No description has been provided for this image

Nos quedamos con la primera diferencia estacional, porque tiene una mejor forma su FAC y FACP, además de tener menor varianza.

BUSQUEDA DE MODELOS¶

Lo que se muestra a continuación fue realizado para la serie sin transformar, solamente estandarizada.

PMDARIMA¶

En 02-----Encontrar-los-Mejores-Modelos se utilizá la libreria de pmdarima para intentar diferentes combinaciones de modelos y encontrar sus $\text{AIC}$.

Se obtuvieron los siguientes resultados:

(1)  ARIMA(1,0,0)(0,1,1)[12], AIC=1430.397
(2)  ARIMA(0,0,1)(0,1,1)[12], AIC=1431.266
(3)  ARIMA(1,0,1)(0,1,1)[12], AIC=1432.133
(4)  ARIMA(2,0,0)(0,1,1)[12], AIC=1432.157
(5)  ARIMA(1,0,0)(0,1,2)[12], AIC=1432.368
(6)  ARIMA(1,0,0)(1,1,1)[12], AIC=1432.373
(7)  ARIMA(0,0,2)(0,1,1)[12], AIC=1432.664
(8)  ARIMA(0,0,1)(0,1,2)[12], AIC=1433.240
(9)  ARIMA(0,0,1)(1,1,1)[12], AIC=1433.245
(10) ARIMA(2,0,1)(0,1,1)[12], AIC=1434.127
(11) ARIMA(0,0,0)(0,1,1)[12], AIC=1445.334
(12) ARIMA(1,0,0)(1,1,0)[12], AIC=1523.141
(13) ARIMA(0,0,1)(1,1,0)[12], AIC=1523.219
(14) ARIMA(0,0,1)(0,1,0)[12], AIC=1652.849
(15) ARIMA(1,0,0)(0,1,0)[12], AIC=1652.984
(16) ARIMA(0,0,0)(0,1,0)[12], AIC=1655.671

Ningun modelo cumplió los supuestos. 02-01-Encontrar-Modelos-que-Cumplan

Por la FAC y FACP, se creyó que podía ser un $\text{ARIMA}(1, 0, 1) \times (0, 1, 1, 12)$, pero este modelo, con la prueba de Ljung-Box, tenía dependencia entre los residuos. Ver 03-----Encontrar-Modelos-que-Cumplan, 1 MODELO.

Se propuso aumentar el número de parámetros, se pusieron parámetros "intuitivos", es decir, la precipitación depende de los últimos 5 meses, de los últimos 4 años, y hay residuos en los últimos dos meses, llegando al primer modelo que cumplía la independecia en los RESIDUOS siendo el $\text{ARIMA}(5, 0, 2) \times (4, 1, 0, 12)$. Sin embargo, no era estacionario ni invertible. Ver 03-----Encontrar-Modelos-que-Cumplan, 2 MODELO.

Se fue variando el número de parámetros de los modelos, notando curiosamente, que para esta serie, que los modelos que eran estacionarios eran los que tenían su parte del polinomio autorregresivo completa y no tenían parte autorregresiva estacional (y viceversa), lo mismo para la parte media móvil y la invertibilidad. Tomando en cuenta esta observación, se hicieron las notebooks de 03-01, ..., 03-04, donde se varían, por ejemplo, los $p$, manteniendo $P=0$, lo mismo para la parte media móvil. Lo malo, es que para los modelos tuvieran independencia en los residuos, los polinomios tenían que ser bastante grandes, lo que podía hacer que no se cumpliera el principio de parsimonia.

En las notebooks 03-01, ..., 03-04 se proponen multiples modelos, que, dada la Observación (2), que dice que para esta serie los modelos que cumplen esa forma son admisibles, se siguió el siguiente procedimiento:

  • Se fijan los valores de $\text{ARIMA}(p, 0, q) \times (0, 1, 0, 12)$ donde $p = 0, 1, 2, \dots$ y $q = 0, 1, 2, \dots$.
  • Se fijan los valores de $\text{ARIMA}(0, 0, q) \times (P, 1, 0, 12)$ donde $P = 0, 1, 2, \dots$ y $q = 0, 1, 2, \dots$.
  • Se fijan los valores de $\text{ARIMA}(p, 0, 0) \times (0, 1, Q, 12)$ donde $p = 0, 1, 2, \dots$ y $Q = 0, 1, 2, \dots$.
  • Se fijan los valores de $\text{ARIMA}(0, 0, 0) \times (P, 1, Q, 12)$ donde $P = 0, 1, 2, \dots$ y $Q = 0, 1, 2, \dots$.

RESULTADOS de 03-01-Encontrar-Modelo-pq¶

(1)   ARIMA(0,0,15)(0,1,0)[12]    AIC=1370.81  
(2)   ARIMA(0,0,12)(0,1,0)[12]    AIC=1370.92  
(3)   ARIMA(3,0,12)(0,1,0)[12]    AIC=1372.25  
(4)   ARIMA(0,0,16)(0,1,0)[12]    AIC=1372.52  
(5)   ARIMA(1,0,15)(0,1,0)[12]    AIC=1372.60  
(6)   ARIMA(1,0,12)(0,1,0)[12]    AIC=1372.69  
(7)   ARIMA(0,0,13)(0,1,0)[12]    AIC=1372.78  
(8)   ARIMA(3,0,13)(0,1,0)[12]    AIC=1372.87  
(9)   ARIMA(4,0,12)(0,1,0)[12]    AIC=1373.41  
(10)  ARIMA(1,0,16)(0,1,0)[12]    AIC=1373.59  
(11)  ARIMA(1,0,13)(0,1,0)[12]    AIC=1373.76  
(12)  ARIMA(3,0,14)(0,1,0)[12]    AIC=1374.01  
(13)  ARIMA(4,0,13)(0,1,0)[12]    AIC=1374.12  
(14)  ARIMA(5,0,12)(0,1,0)[12]    AIC=1374.34  
(15)  ARIMA(0,0,14)(0,1,0)[12]    AIC=1374.44  
(16)  ARIMA(2,0,12)(0,1,0)[12]    AIC=1374.49  
(17)  ARIMA(2,0,12)(0,1,0)[12]    AIC=1374.49  
(18)  ARIMA(0,0,17)(0,1,0)[12]    AIC=1374.49  
(19)  ARIMA(2,0,15)(0,1,0)[12]    AIC=1374.62  
(20)  ARIMA(2,0,15)(0,1,0)[12]    AIC=1374.62  
(21)  ARIMA(1,0,17)(0,1,0)[12]    AIC=1374.97  
(22)  ARIMA(4,0,14)(0,1,0)[12]    AIC=1375.08  
(23)  ARIMA(2,0,13)(0,1,0)[12]    AIC=1375.16  
(24)  ARIMA(2,0,13)(0,1,0)[12]    AIC=1375.16  
(25)  ARIMA(1,0,14)(0,1,0)[12]    AIC=1375.26  
(26)  ARIMA(5,0,13)(0,1,0)[12]    AIC=1375.43  
(27)  ARIMA(6,0,12)(0,1,0)[12]    AIC=1375.94  
(28)  ARIMA(6,0,12)(0,1,0)[12]    AIC=1375.94  
(29)  ARIMA(2,0,16)(0,1,0)[12]    AIC=1375.97  
(30)  ARIMA(2,0,16)(0,1,0)[12]    AIC=1375.97  
(31)  ARIMA(3,0,15)(0,1,0)[12]    AIC=1376.37  
(32)  ARIMA(7,0,12)(0,1,0)[12]    AIC=1376.46  
(33)  ARIMA(2,0,14)(0,1,0)[12]    AIC=1376.63  
(34)  ARIMA(2,0,14)(0,1,0)[12]    AIC=1376.63  
(35)  ARIMA(5,0,14)(0,1,0)[12]    AIC=1377.04  
(36)  ARIMA(6,0,13)(0,1,0)[12]    AIC=1377.52  
(37)  ARIMA(6,0,13)(0,1,0)[12]    AIC=1377.52  
(38)  ARIMA(4,0,15)(0,1,0)[12]    AIC=1377.61  
(39)  ARIMA(2,0,17)(0,1,0)[12]    AIC=1377.67  
(40)  ARIMA(2,0,17)(0,1,0)[12]    AIC=1377.67  
(41)  ARIMA(3,0,16)(0,1,0)[12]    AIC=1377.71  
(42)  ARIMA(7,0,13)(0,1,0)[12]    AIC=1378.15  
(43)  ARIMA(8,0,12)(0,1,0)[12]    AIC=1378.47  
(44)  ARIMA(8,0,12)(0,1,0)[12]    AIC=1378.47  
(45)  ARIMA(9,0,12)(0,1,0)[12]    AIC=1378.79  
(46)  ARIMA(6,0,14)(0,1,0)[12]    AIC=1378.97  
(47)  ARIMA(6,0,14)(0,1,0)[12]    AIC=1378.97  
(48)  ARIMA(4,0,16)(0,1,0)[12]    AIC=1379.46  
(49)  ARIMA(3,0,17)(0,1,0)[12]    AIC=1379.51  
(50)  ARIMA(5,0,15)(0,1,0)[12]    AIC=1379.68  
(51)  ARIMA(8,0,13)(0,1,0)[12]    AIC=1380.15  
(52)  ARIMA(8,0,13)(0,1,0)[12]    AIC=1380.15  
(53)  ARIMA(7,0,14)(0,1,0)[12]    AIC=1380.53  
(54)  ARIMA(10,0,12)(0,1,0)[12]   AIC=1380.62  
(55)  ARIMA(5,0,16)(0,1,0)[12]    AIC=1380.90  
(56)  ARIMA(4,0,17)(0,1,0)[12]    AIC=1381.04  
(57)  ARIMA(6,0,15)(0,1,0)[12]    AIC=1381.49  
(58)  ARIMA(6,0,15)(0,1,0)[12]    AIC=1381.49  
(59)  ARIMA(10,0,13)(0,1,0)[12]   AIC=1381.66  
(60)  ARIMA(9,0,13)(0,1,0)[12]    AIC=1381.75  
(61)  ARIMA(8,0,14)(0,1,0)[12]    AIC=1382.31  
(62)  ARIMA(8,0,14)(0,1,0)[12]    AIC=1382.31  
(63)  ARIMA(5,0,17)(0,1,0)[12]    AIC=1382.35  
(64)  ARIMA(11,0,12)(0,1,0)[12]   AIC=1382.48  
(65)  ARIMA(9,0,14)(0,1,0)[12]    AIC=1382.58  
(66)  ARIMA(6,0,16)(0,1,0)[12]    AIC=1382.64  
(67)  ARIMA(6,0,16)(0,1,0)[12]    AIC=1382.64  
(68)  ARIMA(7,0,15)(0,1,0)[12]    AIC=1382.73  
(69)  ARIMA(10,0,14)(0,1,0)[12]   AIC=1383.53  
(70)  ARIMA(11,0,13)(0,1,0)[12]   AIC=1383.82  
(71)  ARIMA(6,0,17)(0,1,0)[12]    AIC=1384.31  
(72)  ARIMA(6,0,17)(0,1,0)[12]    AIC=1384.31  
(73)  ARIMA(7,0,16)(0,1,0)[12]    AIC=1384.33  
(74)  ARIMA(8,0,15)(0,1,0)[12]    AIC=1384.52  
(75)  ARIMA(8,0,15)(0,1,0)[12]    AIC=1384.52  
(76)  ARIMA(9,0,15)(0,1,0)[12]    AIC=1384.53  
(77)  ARIMA(13,0,12)(0,1,0)[12]   AIC=1384.56  
(78)  ARIMA(11,0,14)(0,1,0)[12]   AIC=1384.73  
(79)  ARIMA(13,0,13)(0,1,0)[12]   AIC=1385.12  
(80)  ARIMA(10,0,15)(0,1,0)[12]   AIC=1385.45  
(81)  ARIMA(12,0,13)(0,1,0)[12]   AIC=1385.52  
(82)  ARIMA(8,0,16)(0,1,0)[12]    AIC=1385.88  
(83)  ARIMA(8,0,16)(0,1,0)[12]    AIC=1385.88  
(84)  ARIMA(12,0,12)(0,1,0)[12]   AIC=1386.15  
(85)  ARIMA(7,0,17)(0,1,0)[12]    AIC=1386.37  
(86)  ARIMA(11,0,15)(0,1,0)[12]   AIC=1386.42  
(87)  ARIMA(9,0,16)(0,1,0)[12]    AIC=1386.43  
(88)  ARIMA(13,0,14)(0,1,0)[12]   AIC=1386.89  
(89)  ARIMA(12,0,14)(0,1,0)[12]   AIC=1386.90  
(90)  ARIMA(10,0,16)(0,1,0)[12]   AIC=1387.11  
(91)  ARIMA(14,0,13)(0,1,0)[12]   AIC=1387.36  
(92)  ARIMA(14,0,12)(0,1,0)[12]   AIC=1387.85  
(93)  ARIMA(11,0,16)(0,1,0)[12]   AIC=1387.97  
(94)  ARIMA(8,0,17)(0,1,0)[12]    AIC=1388.02  
(95)  ARIMA(8,0,17)(0,1,0)[12]    AIC=1388.02  
(96)  ARIMA(15,0,12)(0,1,0)[12]   AIC=1388.42  
(97)  ARIMA(13,0,15)(0,1,0)[12]   AIC=1388.76  
(98)  ARIMA(12,0,15)(0,1,0)[12]   AIC=1388.84  
(99)  ARIMA(14,0,14)(0,1,0)[12]   AIC=1388.89  
(100) ARIMA(9,0,17)(0,1,0)[12]    AIC=1388.95  
(101) ARIMA(10,0,17)(0,1,0)[12]   AIC=1389.04  
(102) ARIMA(15,0,13)(0,1,0)[12]   AIC=1389.30  
(103) ARIMA(17,0,12)(0,1,0)[12]   AIC=1389.57  
(104) ARIMA(11,0,17)(0,1,0)[12]   AIC=1389.72  
(105) ARIMA(16,0,12)(0,1,0)[12]   AIC=1389.87  
(106) ARIMA(12,0,16)(0,1,0)[12]   AIC=1390.33  
(107) ARIMA(15,0,14)(0,1,0)[12]   AIC=1390.73  
(108) ARIMA(14,0,15)(0,1,0)[12]   AIC=1390.78  
(109) ARIMA(17,0,13)(0,1,0)[12]   AIC=1390.81  
(110) ARIMA(16,0,13)(0,1,0)[12]   AIC=1390.83  
(111) ARIMA(13,0,16)(0,1,0)[12]   AIC=1391.95  
(112) ARIMA(12,0,17)(0,1,0)[12]   AIC=1392.26  
(113) ARIMA(13,0,17)(0,1,0)[12]   AIC=1392.28  
(114) ARIMA(17,0,14)(0,1,0)[12]   AIC=1392.49  
(115) ARIMA(16,0,14)(0,1,0)[12]   AIC=1392.69  
(116) ARIMA(15,0,15)(0,1,0)[12]   AIC=1392.85  
(117) ARIMA(14,0,16)(0,1,0)[12]   AIC=1392.97  
(118) ARIMA(16,0,15)(0,1,0)[12]   AIC=1394.72  
(119) ARIMA(17,0,15)(0,1,0)[12]   AIC=1394.84  
(120) ARIMA(15,0,16)(0,1,0)[12]   AIC=1395.10  
(121) ARIMA(14,0,17)(0,1,0)[12]   AIC=1395.38  
(122) ARIMA(15,0,17)(0,1,0)[12]   AIC=1396.22  
(123) ARIMA(17,0,16)(0,1,0)[12]   AIC=1396.75  
(124) ARIMA(16,0,16)(0,1,0)[12]   AIC=1397.15  
(125) ARIMA(16,0,17)(0,1,0)[12]   AIC=1398.28  
(126) ARIMA(17,0,17)(0,1,0)[12]   AIC=1399.30  
(127) ARIMA(9,0,11)(0,1,0)[12]    AIC=1405.22  
(128) ARIMA(11,0,10)(0,1,0)[12]   AIC=1405.63  
(129) ARIMA(11,0,11)(0,1,0)[12]   AIC=1406.36  
(130) ARIMA(10,0,11)(0,1,0)[12]   AIC=1411.83  
(131) ARIMA(14,0,11)(0,1,0)[12]   AIC=1413.61  
(132) ARIMA(13,0,11)(0,1,0)[12]   AIC=1415.15  
(133) ARIMA(8,0,11)(0,1,0)[12]    AIC=1415.83  
(134) ARIMA(8,0,11)(0,1,0)[12]    AIC=1415.83  
(135) ARIMA(15,0,11)(0,1,0)[12]   AIC=1418.52  
(136) ARIMA(16,0,11)(0,1,0)[12]   AIC=1418.96  
(137) ARIMA(12,0,11)(0,1,0)[12]   AIC=1419.07  
(138) ARIMA(7,0,11)(0,1,0)[12]    AIC=1420.16  
(139) ARIMA(15,0,10)(0,1,0)[12]   AIC=1427.03  
(140) ARIMA(16,0,10)(0,1,0)[12]   AIC=1428.10  
(141) ARIMA(16,0,10)(0,1,0)[12]   AIC=1428.10  
(142) ARIMA(14,0,10)(0,1,0)[12]   AIC=1429.08  
(143) ARIMA(4,0,11)(0,1,0)[12]    AIC=1430.74  
(144) ARIMA(3,0,11)(0,1,0)[12]    AIC=1435.07  
(145) ARIMA(17,0,10)(0,1,0)[12]   AIC=1436.99  
(146) ARIMA(12,0,10)(0,1,0)[12]   AIC=1437.18  
(147) ARIMA(5,0,11)(0,1,0)[12]    AIC=1443.88  
(148) ARIMA(13,0,10)(0,1,0)[12]   AIC=1444.13  
(149) ARIMA(6,0,11)(0,1,0)[12]    AIC=1449.08  
(150) ARIMA(6,0,11)(0,1,0)[12]    AIC=1449.08  
(151) ARIMA(16,0,9)(0,1,0)[12]    AIC=1450.39  
(152) ARIMA(17,0,9)(0,1,0)[12]    AIC=1450.46  
(153) ARIMA(2,0,11)(0,1,0)[12]    AIC=1451.97  
(154) ARIMA(2,0,11)(0,1,0)[12]    AIC=1451.97  
(155) ARIMA(16,0,8)(0,1,0)[12]    AIC=1460.41  
(156) ARIMA(17,0,8)(0,1,0)[12]    AIC=1464.61  
(157) ARIMA(16,0,7)(0,1,0)[12]    AIC=1469.14  
(158) ARIMA(17,0,7)(0,1,0)[12]    AIC=1469.22  
(159) ARIMA(17,0,6)(0,1,0)[12]    AIC=1471.02  
(160) ARIMA(9,0,10)(0,1,0)[12]    AIC=1471.14  
(161) ARIMA(8,0,10)(0,1,0)[12]    AIC=1478.49  
(162) ARIMA(8,0,10)(0,1,0)[12]    AIC=1478.49  
(163) ARIMA(16,0,5)(0,1,0)[12]    AIC=1479.55  
(164) ARIMA(16,0,6)(0,1,0)[12]    AIC=1480.98  
(165) ARIMA(10,0,10)(0,1,0)[12]   AIC=1495.41  
(166) ARIMA(17,0,5)(0,1,0)[12]    AIC=1496.61  
(167) ARIMA(6,0,10)(0,1,0)[12]    AIC=1497.53  
(168) ARIMA(6,0,10)(0,1,0)[12]    AIC=1497.53  
(169) ARIMA(1,0,11)(0,1,0)[12]    AIC=1503.08  
(170) ARIMA(5,0,10)(0,1,0)[12]    AIC=1526.92  
(171) ARIMA(10,0,6)(0,1,0)[12]    AIC=1541.36  
(172) ARIMA(4,0,10)(0,1,0)[12]    AIC=1542.28  
(173) ARIMA(7,0,10)(0,1,0)[12]    AIC=1546.44  
(174) ARIMA(3,0,10)(0,1,0)[12]    AIC=1552.06  
(175) ARIMA(2,0,10)(0,1,0)[12]    AIC=1552.72  
(176) ARIMA(2,0,10)(0,1,0)[12]    AIC=1552.72  
(177) ARIMA(2,0,10)(0,1,0)[12]    AIC=1552.72  
(178) ARIMA(2,0,9)(0,1,0)[12]     AIC=1555.97  
(179) ARIMA(2,0,8)(0,1,0)[12]     AIC=1559.09  
(180) ARIMA(1,0,8)(0,1,0)[12]     AIC=1593.01  
(181) ARIMA(0,0,11)(0,1,0)[12]    AIC=1599.58  
(182) ARIMA(2,0,5)(0,1,0)[12]     AIC=1601.11  
(183) ARIMA(2,0,6)(0,1,0)[12]     AIC=1621.74  
(184) ARIMA(0,0,9)(0,1,0)[12]     AIC=1621.79  
(185) ARIMA(0,0,10)(0,1,0)[12]    AIC=1622.73  
(186) ARIMA(0,0,10)(0,1,0)[12]    AIC=1622.73  
(187) ARIMA(1,0,9)(0,1,0)[12]     AIC=1623.45  
(188) ARIMA(1,0,10)(0,1,0)[12]    AIC=1624.20  
(189) ARIMA(1,0,10)(0,1,0)[12]    AIC=1624.20  
(190) ARIMA(1,0,5)(0,1,0)[12]     AIC=1627.56  
(191) ARIMA(1,0,6)(0,1,0)[12]     AIC=1630.49  
(192) ARIMA(0,0,7)(0,1,0)[12]     AIC=1630.59  
(193) ARIMA(0,0,8)(0,1,0)[12]     AIC=1630.74  
(194) ARIMA(1,0,7)(0,1,0)[12]     AIC=1631.92  
(195) ARIMA(2,0,7)(0,1,0)[12]     AIC=1633.08  
(196) ARIMA(0,0,6)(0,1,0)[12]     AIC=1635.97  
(197) ARIMA(0,0,5)(0,1,0)[12]     AIC=1651.91

RESULTADOS de 03-02-Encontrar-Modelo-Pq¶

(1)   ARIMA(0,0,15)(0,1,0)[12]    AIC=1370.81  
(2)   ARIMA(0,0,12)(0,1,0)[12]    AIC=1370.92  
(3)   ARIMA(0,0,12)(1,1,0)[12]    AIC=1372.79  
(4)   ARIMA(0,0,15)(1,1,0)[12]    AIC=1373.17  
(5)   ARIMA(0,0,12)(2,1,0)[12]    AIC=1373.37  
(6)   ARIMA(0,0,13)(1,1,0)[12]    AIC=1373.86  
(7)   ARIMA(0,0,12)(4,1,0)[12]    AIC=1375.93  
(8)   ARIMA(0,0,14)(2,1,0)[12]    AIC=1377.49  
(9)   ARIMA(0,0,11)(3,1,0)[12]    AIC=1427.35  
(10)  ARIMA(0,0,10)(3,1,0)[12]    AIC=1430.51  
(11)  ARIMA(0,0,11)(2,1,0)[12]    AIC=1443.12  
(12)  ARIMA(0,0,10)(2,1,0)[12]    AIC=1446.97  
(13)  ARIMA(0,0,11)(1,1,0)[12]    AIC=1511.21  
(14)  ARIMA(0,0,6)(1,1,0)[12]     AIC=1514.47  
(15)  ARIMA(0,0,8)(1,1,0)[12]     AIC=1516.15  
(16)  ARIMA(0,0,10)(1,1,0)[12]    AIC=1516.98

RESULTADOS de 03-03-Encontrar-Modelo-pQ¶

(1)   ARIMA(12,0,0)(0,1,4)[12]    AIC=1367.04  
(2)   ARIMA(13,0,0)(0,1,4)[12]    AIC=1367.37  
(3)   ARIMA(12,0,0)(0,1,3)[12]    AIC=1368.24  
(4)   ARIMA(12,0,0)(0,1,5)[12]    AIC=1369.25  
(5)   ARIMA(15,0,0)(0,1,4)[12]    AIC=1371.49  
(6)   ARIMA(12,0,0)(0,1,1)[12]    AIC=1382.66  
(7)   ARIMA(15,0,0)(0,1,1)[12]    AIC=1387.68  
(8)   ARIMA(6,0,0)(0,1,1)[12]     AIC=1391.61  
(9)   ARIMA(11,0,0)(0,1,4)[12]    AIC=1392.63  
(10)  ARIMA(10,0,0)(0,1,4)[12]    AIC=1398.90  
(11)  ARIMA(3,0,0)(0,1,1)[12]     AIC=1431.92  
(12)  ARIMA(0,0,0)(0,1,1)[12]     AIC=1443.59  
(13)  ARIMA(12,0,0)(0,1,0)[12]    AIC=1522.91  
(14)  ARIMA(6,0,0)(0,1,0)[12]     AIC=1650.81

RESULTADOS de 03-04-Encontrar-Modelo-PQ¶

(1)   ARIMA(0,0,0)(0,1,3)[12]     AIC=1443.50  
(2)   ARIMA(0,0,0)(0,1,1)[12]     AIC=1443.59  
(3)   ARIMA(0,0,0)(2,1,1)[12]     AIC=1444.11  
(4)   ARIMA(0,0,0)(1,1,2)[12]     AIC=1445.15  
(5)   ARIMA(0,0,0)(1,1,3)[12]     AIC=1445.28  
(6)   ARIMA(0,0,0)(0,1,4)[12]     AIC=1445.49  
(7)   ARIMA(0,0,0)(0,1,2)[12]     AIC=1445.59  
(8)   ARIMA(0,0,0)(1,1,1)[12]     AIC=1445.59  
(9)   ARIMA(0,0,0)(4,1,3)[12]     AIC=1445.70  
(10)  ARIMA(0,0,0)(4,1,4)[12]     AIC=1445.74  
(11)  ARIMA(0,0,0)(2,1,2)[12]     AIC=1445.92  
(12)  ARIMA(0,0,0)(4,1,2)[12]     AIC=1445.95  
(13)  ARIMA(0,0,0)(3,1,1)[12]     AIC=1446.07  
(14)  ARIMA(0,0,0)(2,1,3)[12]     AIC=1446.38  
(15)  ARIMA(0,0,0)(1,1,4)[12]     AIC=1446.57  
(16)  ARIMA(0,0,0)(4,1,1)[12]     AIC=1446.70  
(17)  ARIMA(0,0,0)(3,1,2)[12]     AIC=1446.99  
(18)  ARIMA(0,0,0)(3,1,3)[12]     AIC=1447.78  
(19)  ARIMA(0,0,0)(2,1,4)[12]     AIC=1447.80  
(20)  ARIMA(0,0,0)(4,1,0)[12]     AIC=1448.24  
(21)  ARIMA(0,0,0)(3,1,4)[12]     AIC=1449.21  
(22)  ARIMA(0,0,0)(3,1,0)[12]     AIC=1454.13  
(23)  ARIMA(0,0,0)(2,1,0)[12]     AIC=1464.14  
(24)  ARIMA(0,0,0)(1,1,0)[12]     AIC=1526.64  
(25)  ARIMA(0,0,0)(0,1,0)[12]     AIC=1653.67

MODELOS ORDENADOS POR SU $\text{AIC}$:¶

Estos son todos los modelos propuestos, ordenados por su AIC:

(1) ARIMA(12,0,0)(0,1,4)[12], AIC=1367.04
(2) ARIMA(13,0,0)(0,1,4)[12], AIC=1367.37
(3) ARIMA(12,0,0)(0,1,3)[12], AIC=1368.24
(4) ARIMA(12,0,0)(0,1,5)[12], AIC=1369.25
(5) ARIMA(0,0,15)(0,1,0)[12], AIC=1370.81
(6) ARIMA(0,0,15)(0,1,0)[12], AIC=1370.81
(7) ARIMA(0,0,12)(0,1,0)[12], AIC=1370.92
(8) ARIMA(0,0,12)(0,1,0)[12], AIC=1370.92
(9) ARIMA(15,0,0)(0,1,4)[12], AIC=1371.49
(10) ARIMA(3,0,12)(0,1,0)[12], AIC=1372.25
(11) ARIMA(0,0,16)(0,1,0)[12], AIC=1372.52
(12) ARIMA(1,0,15)(0,1,0)[12], AIC=1372.60
(13) ARIMA(1,0,12)(0,1,0)[12], AIC=1372.69
(14) ARIMA(0,0,13)(0,1,0)[12], AIC=1372.78
(15) ARIMA(0,0,12)(1,1,0)[12], AIC=1372.79
(16) ARIMA(3,0,13)(0,1,0)[12], AIC=1372.87
(17) ARIMA(0,0,15)(1,1,0)[12], AIC=1373.17
(18) ARIMA(0,0,12)(2,1,0)[12], AIC=1373.37
(19) ARIMA(4,0,12)(0,1,0)[12], AIC=1373.41
(20) ARIMA(1,0,16)(0,1,0)[12], AIC=1373.59
(21) ARIMA(1,0,13)(0,1,0)[12], AIC=1373.76
(22) ARIMA(0,0,13)(1,1,0)[12], AIC=1373.86
(23) ARIMA(3,0,14)(0,1,0)[12], AIC=1374.01
(24) ARIMA(4,0,13)(0,1,0)[12], AIC=1374.12
(25) ARIMA(5,0,12)(0,1,0)[12], AIC=1374.34
(26) ARIMA(0,0,14)(0,1,0)[12], AIC=1374.44
(27) ARIMA(2,0,12)(0,1,0)[12], AIC=1374.49
(28) ARIMA(2,0,12)(0,1,0)[12], AIC=1374.49
(29) ARIMA(0,0,17)(0,1,0)[12], AIC=1374.49
(30) ARIMA(2,0,15)(0,1,0)[12], AIC=1374.62
(31) ARIMA(2,0,15)(0,1,0)[12], AIC=1374.62
(32) ARIMA(1,0,17)(0,1,0)[12], AIC=1374.97
(33) ARIMA(4,0,14)(0,1,0)[12], AIC=1375.08
(34) ARIMA(2,0,13)(0,1,0)[12], AIC=1375.16
(35) ARIMA(2,0,13)(0,1,0)[12], AIC=1375.16
(36) ARIMA(1,0,14)(0,1,0)[12], AIC=1375.26
(37) ARIMA(5,0,13)(0,1,0)[12], AIC=1375.43
(38) ARIMA(0,0,12)(4,1,0)[12], AIC=1375.93
(39) ARIMA(6,0,12)(0,1,0)[12], AIC=1375.94
(40) ARIMA(6,0,12)(0,1,0)[12], AIC=1375.94
(41) ARIMA(2,0,16)(0,1,0)[12], AIC=1375.97
(42) ARIMA(2,0,16)(0,1,0)[12], AIC=1375.97
(43) ARIMA(3,0,15)(0,1,0)[12], AIC=1376.37
(44) ARIMA(7,0,12)(0,1,0)[12], AIC=1376.46
(45) ARIMA(2,0,14)(0,1,0)[12], AIC=1376.63
(46) ARIMA(2,0,14)(0,1,0)[12], AIC=1376.63
(47) ARIMA(5,0,14)(0,1,0)[12], AIC=1377.04
(48) ARIMA(0,0,14)(2,1,0)[12], AIC=1377.49
(49) ARIMA(6,0,13)(0,1,0)[12], AIC=1377.52
(50) ARIMA(6,0,13)(0,1,0)[12], AIC=1377.52
(51) ARIMA(4,0,15)(0,1,0)[12], AIC=1377.61
(52) ARIMA(2,0,17)(0,1,0)[12], AIC=1377.67
(53) ARIMA(2,0,17)(0,1,0)[12], AIC=1377.67
(54) ARIMA(3,0,16)(0,1,0)[12], AIC=1377.71
(55) ARIMA(7,0,13)(0,1,0)[12], AIC=1378.15
(56) ARIMA(8,0,12)(0,1,0)[12], AIC=1378.47
(57) ARIMA(8,0,12)(0,1,0)[12], AIC=1378.47
(58) ARIMA(9,0,12)(0,1,0)[12], AIC=1378.79
(59) ARIMA(6,0,14)(0,1,0)[12], AIC=1378.97
(60) ARIMA(6,0,14)(0,1,0)[12], AIC=1378.97
(61) ARIMA(4,0,16)(0,1,0)[12], AIC=1379.46
(62) ARIMA(3,0,17)(0,1,0)[12], AIC=1379.51
(63) ARIMA(5,0,15)(0,1,0)[12], AIC=1379.68
(64) ARIMA(8,0,13)(0,1,0)[12], AIC=1380.15
(65) ARIMA(8,0,13)(0,1,0)[12], AIC=1380.15
(66) ARIMA(7,0,14)(0,1,0)[12], AIC=1380.53
(67) ARIMA(10,0,12)(0,1,0)[12], AIC=1380.62
(68) ARIMA(5,0,16)(0,1,0)[12], AIC=1380.90
(69) ARIMA(4,0,17)(0,1,0)[12], AIC=1381.04
(70) ARIMA(6,0,15)(0,1,0)[12], AIC=1381.49
(71) ARIMA(6,0,15)(0,1,0)[12], AIC=1381.49
(72) ARIMA(10,0,13)(0,1,0)[12], AIC=1381.66
(73) ARIMA(9,0,13)(0,1,0)[12], AIC=1381.75
(74) ARIMA(8,0,14)(0,1,0)[12], AIC=1382.31
(75) ARIMA(8,0,14)(0,1,0)[12], AIC=1382.31
(76) ARIMA(5,0,17)(0,1,0)[12], AIC=1382.35
(77) ARIMA(11,0,12)(0,1,0)[12], AIC=1382.48
(78) ARIMA(9,0,14)(0,1,0)[12], AIC=1382.58
(79) ARIMA(6,0,16)(0,1,0)[12], AIC=1382.64
(80) ARIMA(6,0,16)(0,1,0)[12], AIC=1382.64
(81) ARIMA(12,0,0)(0,1,1)[12], AIC=1382.66
(82) ARIMA(7,0,15)(0,1,0)[12], AIC=1382.73
(83) ARIMA(10,0,14)(0,1,0)[12], AIC=1383.53
(84) ARIMA(11,0,13)(0,1,0)[12], AIC=1383.82
(85) ARIMA(6,0,17)(0,1,0)[12], AIC=1384.31
(86) ARIMA(6,0,17)(0,1,0)[12], AIC=1384.31
(87) ARIMA(7,0,16)(0,1,0)[12], AIC=1384.33
(88) ARIMA(8,0,15)(0,1,0)[12], AIC=1384.52
(89) ARIMA(8,0,15)(0,1,0)[12], AIC=1384.52
(90) ARIMA(9,0,15)(0,1,0)[12], AIC=1384.53
(91) ARIMA(13,0,12)(0,1,0)[12], AIC=1384.56
(92) ARIMA(11,0,14)(0,1,0)[12], AIC=1384.73
(93) ARIMA(13,0,13)(0,1,0)[12], AIC=1385.12
(94) ARIMA(10,0,15)(0,1,0)[12], AIC=1385.45
(95) ARIMA(12,0,13)(0,1,0)[12], AIC=1385.52
(96) ARIMA(8,0,16)(0,1,0)[12], AIC=1385.88
(97) ARIMA(8,0,16)(0,1,0)[12], AIC=1385.88
(98) ARIMA(12,0,12)(0,1,0)[12], AIC=1386.15
(99) ARIMA(7,0,17)(0,1,0)[12], AIC=1386.37
(100) ARIMA(11,0,15)(0,1,0)[12], AIC=1386.42
(101) ARIMA(9,0,16)(0,1,0)[12], AIC=1386.43
(102) ARIMA(13,0,14)(0,1,0)[12], AIC=1386.89
(103) ARIMA(12,0,14)(0,1,0)[12], AIC=1386.90
(104) ARIMA(10,0,16)(0,1,0)[12], AIC=1387.11
(105) ARIMA(14,0,13)(0,1,0)[12], AIC=1387.36
(106) ARIMA(15,0,0)(0,1,1)[12], AIC=1387.68
(107) ARIMA(14,0,12)(0,1,0)[12], AIC=1387.85
(108) ARIMA(11,0,16)(0,1,0)[12], AIC=1387.97
(109) ARIMA(8,0,17)(0,1,0)[12], AIC=1388.02
(110) ARIMA(8,0,17)(0,1,0)[12], AIC=1388.02
(111) ARIMA(15,0,12)(0,1,0)[12], AIC=1388.42
(112) ARIMA(13,0,15)(0,1,0)[12], AIC=1388.76
(113) ARIMA(12,0,15)(0,1,0)[12], AIC=1388.84
(114) ARIMA(14,0,14)(0,1,0)[12], AIC=1388.89
(115) ARIMA(9,0,17)(0,1,0)[12], AIC=1388.95
(116) ARIMA(10,0,17)(0,1,0)[12], AIC=1389.04
(117) ARIMA(15,0,13)(0,1,0)[12], AIC=1389.30
(118) ARIMA(17,0,12)(0,1,0)[12], AIC=1389.57
(119) ARIMA(11,0,17)(0,1,0)[12], AIC=1389.72
(120) ARIMA(16,0,12)(0,1,0)[12], AIC=1389.87
(121) ARIMA(12,0,16)(0,1,0)[12], AIC=1390.33
(122) ARIMA(15,0,14)(0,1,0)[12], AIC=1390.73
(123) ARIMA(14,0,15)(0,1,0)[12], AIC=1390.78
(124) ARIMA(17,0,13)(0,1,0)[12], AIC=1390.81
(125) ARIMA(16,0,13)(0,1,0)[12], AIC=1390.83
(126) ARIMA(6,0,0)(0,1,1)[12], AIC=1391.61
(127) ARIMA(13,0,16)(0,1,0)[12], AIC=1391.95
(128) ARIMA(12,0,17)(0,1,0)[12], AIC=1392.26
(129) ARIMA(13,0,17)(0,1,0)[12], AIC=1392.28
(130) ARIMA(17,0,14)(0,1,0)[12], AIC=1392.49
(131) ARIMA(11,0,0)(0,1,4)[12], AIC=1392.63
(132) ARIMA(16,0,14)(0,1,0)[12], AIC=1392.69
(133) ARIMA(15,0,15)(0,1,0)[12], AIC=1392.85
(134) ARIMA(14,0,16)(0,1,0)[12], AIC=1392.97
(135) ARIMA(16,0,15)(0,1,0)[12], AIC=1394.72
(136) ARIMA(17,0,15)(0,1,0)[12], AIC=1394.84
(137) ARIMA(15,0,16)(0,1,0)[12], AIC=1395.10
(138) ARIMA(14,0,17)(0,1,0)[12], AIC=1395.38
(139) ARIMA(15,0,17)(0,1,0)[12], AIC=1396.22
(140) ARIMA(17,0,16)(0,1,0)[12], AIC=1396.75
(141) ARIMA(16,0,16)(0,1,0)[12], AIC=1397.15
(142) ARIMA(16,0,17)(0,1,0)[12], AIC=1398.28
(143) ARIMA(10,0,0)(0,1,4)[12], AIC=1398.90
(144) ARIMA(17,0,17)(0,1,0)[12], AIC=1399.30
(145) ARIMA(9,0,11)(0,1,0)[12], AIC=1405.22
(146) ARIMA(11,0,10)(0,1,0)[12], AIC=1405.63
(147) ARIMA(11,0,11)(0,1,0)[12], AIC=1406.36
(148) ARIMA(10,0,11)(0,1,0)[12], AIC=1411.83
(149) ARIMA(14,0,11)(0,1,0)[12], AIC=1413.61
(150) ARIMA(13,0,11)(0,1,0)[12], AIC=1415.15
(151) ARIMA(8,0,11)(0,1,0)[12], AIC=1415.83
(152) ARIMA(8,0,11)(0,1,0)[12], AIC=1415.83
(153) ARIMA(15,0,11)(0,1,0)[12], AIC=1418.52
(154) ARIMA(16,0,11)(0,1,0)[12], AIC=1418.96
(155) ARIMA(12,0,11)(0,1,0)[12], AIC=1419.07
(156) ARIMA(7,0,11)(0,1,0)[12], AIC=1420.16
(157) ARIMA(15,0,10)(0,1,0)[12], AIC=1427.03
(158) ARIMA(0,0,11)(3,1,0)[12], AIC=1427.35
(159) ARIMA(16,0,10)(0,1,0)[12], AIC=1428.10
(160) ARIMA(16,0,10)(0,1,0)[12], AIC=1428.10
(161) ARIMA(14,0,10)(0,1,0)[12], AIC=1429.08
(162) ARIMA(0,0,10)(3,1,0)[12], AIC=1430.51
(163) ARIMA(4,0,11)(0,1,0)[12], AIC=1430.74
(164) ARIMA(3,0,0)(0,1,1)[12], AIC=1431.92
(165) ARIMA(3,0,11)(0,1,0)[12], AIC=1435.07
(166) ARIMA(17,0,10)(0,1,0)[12], AIC=1436.99
(167) ARIMA(12,0,10)(0,1,0)[12], AIC=1437.18
(168) ARIMA(0,0,11)(2,1,0)[12], AIC=1443.12
(169) ARIMA(0,0,0)(0,1,3)[12], AIC=1443.50
(170) ARIMA(0,0,0)(0,1,1)[12], AIC=1443.59
(171) ARIMA(0,0,0)(0,1,1)[12], AIC=1443.59
(172) ARIMA(5,0,11)(0,1,0)[12], AIC=1443.88
(173) ARIMA(0,0,0)(2,1,1)[12], AIC=1444.11
(174) ARIMA(13,0,10)(0,1,0)[12], AIC=1444.13
(175) ARIMA(0,0,0)(1,1,2)[12], AIC=1445.15
(176) ARIMA(0,0,0)(1,1,3)[12], AIC=1445.28
(177) ARIMA(0,0,0)(0,1,4)[12], AIC=1445.49
(178) ARIMA(0,0,0)(0,1,2)[12], AIC=1445.59
(179) ARIMA(0,0,0)(1,1,1)[12], AIC=1445.59
(180) ARIMA(0,0,0)(4,1,3)[12], AIC=1445.70
(181) ARIMA(0,0,0)(4,1,4)[12], AIC=1445.74
(182) ARIMA(0,0,0)(2,1,2)[12], AIC=1445.92
(183) ARIMA(0,0,0)(4,1,2)[12], AIC=1445.95
(184) ARIMA(0,0,0)(3,1,1)[12], AIC=1446.07
(185) ARIMA(0,0,0)(2,1,3)[12], AIC=1446.38
(186) ARIMA(0,0,0)(1,1,4)[12], AIC=1446.57
(187) ARIMA(0,0,0)(4,1,1)[12], AIC=1446.70
(188) ARIMA(0,0,10)(2,1,0)[12], AIC=1446.97
(189) ARIMA(0,0,0)(3,1,2)[12], AIC=1446.99
(190) ARIMA(0,0,0)(3,1,3)[12], AIC=1447.78
(191) ARIMA(0,0,0)(2,1,4)[12], AIC=1447.80
(192) ARIMA(0,0,0)(4,1,0)[12], AIC=1448.24
(193) ARIMA(6,0,11)(0,1,0)[12], AIC=1449.08
(194) ARIMA(6,0,11)(0,1,0)[12], AIC=1449.08
(195) ARIMA(0,0,0)(3,1,4)[12], AIC=1449.21
(196) ARIMA(16,0,9)(0,1,0)[12], AIC=1450.39
(197) ARIMA(17,0,9)(0,1,0)[12], AIC=1450.46
(198) ARIMA(2,0,11)(0,1,0)[12], AIC=1451.97
(199) ARIMA(2,0,11)(0,1,0)[12], AIC=1451.97
(200) ARIMA(0,0,0)(3,1,0)[12], AIC=1454.13
(201) ARIMA(16,0,8)(0,1,0)[12], AIC=1460.41
(202) ARIMA(0,0,0)(2,1,0)[12], AIC=1464.14
(203) ARIMA(17,0,8)(0,1,0)[12], AIC=1464.61
(204) ARIMA(16,0,7)(0,1,0)[12], AIC=1469.14
(205) ARIMA(17,0,7)(0,1,0)[12], AIC=1469.22
(206) ARIMA(17,0,6)(0,1,0)[12], AIC=1471.02
(207) ARIMA(9,0,10)(0,1,0)[12], AIC=1471.14
(208) ARIMA(8,0,10)(0,1,0)[12], AIC=1478.49
(209) ARIMA(8,0,10)(0,1,0)[12], AIC=1478.49
(210) ARIMA(16,0,5)(0,1,0)[12], AIC=1479.55
(211) ARIMA(16,0,6)(0,1,0)[12], AIC=1480.98
(212) ARIMA(10,0,10)(0,1,0)[12], AIC=1495.41
(213) ARIMA(17,0,5)(0,1,0)[12], AIC=1496.61
(214) ARIMA(6,0,10)(0,1,0)[12], AIC=1497.53
(215) ARIMA(6,0,10)(0,1,0)[12], AIC=1497.53
(216) ARIMA(1,0,11)(0,1,0)[12], AIC=1503.08
(217) ARIMA(0,0,11)(1,1,0)[12], AIC=1511.21
(218) ARIMA(0,0,6)(1,1,0)[12], AIC=1514.47
(219) ARIMA(0,0,8)(1,1,0)[12], AIC=1516.15
(220) ARIMA(0,0,10)(1,1,0)[12], AIC=1516.98
(221) ARIMA(12,0,0)(0,1,0)[12], AIC=1522.91
(222) ARIMA(0,0,0)(1,1,0)[12], AIC=1526.64
(223) ARIMA(5,0,10)(0,1,0)[12], AIC=1526.92
(224) ARIMA(10,0,6)(0,1,0)[12], AIC=1541.36
(225) ARIMA(4,0,10)(0,1,0)[12], AIC=1542.28
(226) ARIMA(7,0,10)(0,1,0)[12], AIC=1546.44
(227) ARIMA(3,0,10)(0,1,0)[12], AIC=1552.06
(228) ARIMA(2,0,10)(0,1,0)[12], AIC=1552.72
(229) ARIMA(2,0,10)(0,1,0)[12], AIC=1552.72
(230) ARIMA(2,0,10)(0,1,0)[12], AIC=1552.72
(231) ARIMA(2,0,9)(0,1,0)[12], AIC=1555.97
(232) ARIMA(2,0,8)(0,1,0)[12], AIC=1559.09
(233) ARIMA(1,0,8)(0,1,0)[12], AIC=1593.01
(234) ARIMA(0,0,11)(0,1,0)[12], AIC=1599.58
(235) ARIMA(2,0,5)(0,1,0)[12], AIC=1601.11
(236) ARIMA(2,0,6)(0,1,0)[12], AIC=1621.74
(237) ARIMA(0,0,9)(0,1,0)[12], AIC=1621.79
(238) ARIMA(0,0,10)(0,1,0)[12], AIC=1622.73
(239) ARIMA(0,0,10)(0,1,0)[12], AIC=1622.73
(240) ARIMA(1,0,9)(0,1,0)[12], AIC=1623.45
(241) ARIMA(1,0,10)(0,1,0)[12], AIC=1624.20
(242) ARIMA(1,0,10)(0,1,0)[12], AIC=1624.20
(243) ARIMA(1,0,5)(0,1,0)[12], AIC=1627.56
(244) ARIMA(1,0,6)(0,1,0)[12], AIC=1630.49
(245) ARIMA(0,0,7)(0,1,0)[12], AIC=1630.59
(246) ARIMA(0,0,8)(0,1,0)[12], AIC=1630.74
(247) ARIMA(1,0,7)(0,1,0)[12], AIC=1631.92
(248) ARIMA(2,0,7)(0,1,0)[12], AIC=1633.08
(249) ARIMA(0,0,6)(0,1,0)[12], AIC=1635.97
(250) ARIMA(6,0,0)(0,1,0)[12], AIC=1650.81
(251) ARIMA(0,0,5)(0,1,0)[12], AIC=1651.91
(252) ARIMA(0,0,0)(0,1,0)[12], AIC=1653.67

MODELOS PROPUESTOS¶

MODELOS PARA LA SERIE SIN TRANSFORMAR¶

PRIMER MODELO¶

SARIMAX Results
Dep. Variable: y No. Observations: 660
Model: SARIMAX(12, 0, 0)x(0, 1, [1, 2, 3, 4], 12) Log Likelihood -666.522
Date: Sun, 27 Apr 2025 AIC 1367.044
Time: 04:48:12 BIC 1443.100
Sample: 0 HQIC 1396.549
- 660
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 0.1079 0.032 3.338 0.001 0.045 0.171
ar.L2 -0.0271 0.037 -0.738 0.460 -0.099 0.045
ar.L3 0.0504 0.037 1.367 0.172 -0.022 0.123
ar.L4 -0.0689 0.042 -1.634 0.102 -0.152 0.014
ar.L5 -0.1086 0.050 -2.193 0.028 -0.206 -0.012
ar.L6 -0.1103 0.059 -1.872 0.061 -0.226 0.005
ar.L7 -0.0049 0.050 -0.097 0.923 -0.104 0.094
ar.L8 -0.0450 0.043 -1.039 0.299 -0.130 0.040
ar.L9 0.0411 0.035 1.183 0.237 -0.027 0.109
ar.L10 -0.0133 0.033 -0.406 0.685 -0.078 0.051
ar.L11 0.0968 0.036 2.662 0.008 0.026 0.168
ar.L12 0.5442 0.074 7.328 0.000 0.399 0.690
ma.S.L12 -1.3768 0.084 -16.419 0.000 -1.541 -1.212
ma.S.L24 0.3656 0.079 4.639 0.000 0.211 0.520
ma.S.L36 0.1054 0.060 1.756 0.079 -0.012 0.223
ma.S.L48 -0.0790 0.039 -2.042 0.041 -0.155 -0.003
sigma2 0.4378 0.024 18.022 0.000 0.390 0.485
Ljung-Box (L1) (Q): 0.34 Jarque-Bera (JB): 376.73
Prob(Q): 0.56 Prob(JB): 0.00
Heteroskedasticity (H): 0.82 Skew: 0.89
Prob(H) (two-sided): 0.14 Kurtosis: 6.28


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

VERIFICACIÓN DE SUPUESTOS¶

PRINCIPIO DE PARSIMONIA¶
Hay coeficientes no significativos, no se cumple el principio de parsimonia
MODELO ADMISIBLE¶
El polinomio de la parte autoregresiva es: {'p1': -0.10787169558218329, 'p2': 0.027085643770389415, 'p3': -0.050379964730704004, 'p4': 0.06894551834156623, 'p5': 0.10861700542171245, 'p6': 0.11029529000485752, 'p7': 0.00488096912398539, 'p8': 0.04497037748451452, 'p9': -0.04112819961848579, 'p10': 0.013332216884030874, 'p11': -0.09678291912170793, 'p12': -0.5442160802231802}

El grado del polinomio es: 12

Raíces del polinomio característico: [-1.10915274+0.j         -0.92789955+0.52544644j -0.92789955-0.52544644j
 -0.52132244+0.91355779j -0.52132244-0.91355779j -0.01548825+1.04638174j
 -0.01548825-1.04638174j  0.53546751+0.91668064j  0.53546751-0.91668064j
  1.05996383+0.j          0.86491762+0.50841197j  0.86491762-0.50841197j]

Módulo de las raíces: [1.10915274 1.06634494 1.06634494 1.05183883 1.05183883 1.04649636
 1.04649636 1.06161615 1.06161615 1.05996383 1.00327724 1.00327724]

¿Las raíces están fuera del círculo unitario?  True

El modelo es estacionario

:)
El polinomio de la parte media móvil es: {'p12': -1.376803188196061, 'p24': 0.36563094250781014, 'p36': 0.10536384747112354, 'p48': -0.07904343456059729}

El grado del polinomio es: 48

Raíces del polinomio característico: [-1.05513205e+00+0.28272178j -1.05513205e+00-0.28272178j
 -1.06072484e+00+0.06092667j -1.06072484e+00-0.06092667j
 -1.00194881e+00+0.j         -7.72410271e-01+0.77241027j
 -7.72410271e-01-0.77241027j -9.49077989e-01+0.47759838j
 -9.49077989e-01-0.47759838j -8.88151320e-01+0.58312646j
 -8.88151320e-01-0.58312646j -8.67713125e-01+0.50097441j
 -8.67713125e-01-0.50097441j -5.83126461e-01+0.88815132j
 -5.83126461e-01-0.88815132j -4.77598375e-01+0.94907799j
 -4.77598375e-01-0.94907799j -5.00974406e-01+0.86771313j
 -5.00974406e-01-0.86771313j -2.82721781e-01+1.05513205j
 -2.82721781e-01-1.05513205j -6.09266684e-02+1.06072484j
 -6.09266684e-02-1.06072484j  6.09266684e-02+1.06072484j
  6.09266684e-02-1.06072484j  4.53730117e-17+1.00194881j
  4.53730117e-17-1.00194881j  2.82721781e-01+1.05513205j
  2.82721781e-01-1.05513205j  4.77598375e-01+0.94907799j
  4.77598375e-01-0.94907799j  5.83126461e-01+0.88815132j
  5.83126461e-01-0.88815132j  5.00974406e-01+0.86771313j
  5.00974406e-01-0.86771313j  7.72410271e-01+0.77241027j
  7.72410271e-01-0.77241027j  1.05513205e+00+0.28272178j
  1.05513205e+00-0.28272178j  1.06072484e+00+0.06092667j
  1.06072484e+00-0.06092667j  1.00194881e+00+0.j
  8.88151320e-01+0.58312646j  8.88151320e-01-0.58312646j
  9.49077989e-01+0.47759838j  9.49077989e-01-0.47759838j
  8.67713125e-01+0.50097441j  8.67713125e-01-0.50097441j]

Módulo de las raíces: [1.09235308 1.09235308 1.06247317 1.06247317 1.00194881 1.09235308
 1.09235308 1.06247317 1.06247317 1.06247317 1.06247317 1.00194881
 1.00194881 1.06247317 1.06247317 1.06247317 1.06247317 1.00194881
 1.00194881 1.09235308 1.09235308 1.06247317 1.06247317 1.06247317
 1.06247317 1.00194881 1.00194881 1.09235308 1.09235308 1.06247317
 1.06247317 1.06247317 1.06247317 1.00194881 1.00194881 1.09235308
 1.09235308 1.09235308 1.09235308 1.06247317 1.06247317 1.00194881
 1.06247317 1.06247317 1.06247317 1.06247317 1.00194881 1.00194881]

¿Las raíces están fuera del círculo unitario?  True

El modelo es invertible

:)
RESIDUOS INDEPENDIENTES¶
lb_stat lb_pvalue
1 1.410877 NaN
2 1.666736 NaN
3 1.668529 NaN
4 2.153928 NaN
5 3.756515 NaN
6 4.271537 NaN
7 4.271537 NaN
8 4.398558 NaN
9 4.673186 NaN
10 5.854609 NaN
11 6.235510 NaN
12 6.364705 NaN
13 8.908873 NaN
14 9.013624 NaN
15 9.115022 NaN
16 9.301973 NaN
17 9.622827 0.001922
18 9.634894 0.008087
19 9.648719 0.021801
20 9.650120 0.046752
21 10.177061 0.070371
22 10.403762 0.108646
23 10.403778 0.166823
24 10.406790 0.237627
25 10.406958 0.318555
26 10.956519 0.360916
27 11.577235 0.396244
28 11.768293 0.464463
29 12.219944 0.509696
30 12.223270 0.588378
31 12.229827 0.661556
32 14.130904 0.588962
33 14.189628 0.653636
34 16.160520 0.581346
35 16.186598 0.644794
36 16.663770 0.674686
37 16.699639 0.729145
38 17.046283 0.760802
39 17.392114 0.789577
40 18.027704 0.801662
41 19.922150 0.750897
42 20.004090 0.791363
43 20.038756 0.829126
44 20.096537 0.860920
45 20.910376 0.862394
46 21.002354 0.887814
47 21.052280 0.910525
48 21.957194 0.908534

Hay dependencia en los primeros dos residuos.

RESIDUOS CON MEDIA CERO¶
-0.04082317526120878
TtestResult(statistic=-1.5347957438291504, pvalue=0.12531394692788528, df=659)

La media de los residuos puede ser 0.

RESIDUOS CON VARIANZA CONSTANTE¶
(2.3754715860699527,
 0.12325437239232831,
 2.376827864684162,
 0.12362813746633024)

Los residuos tienen varianza constante.

RESIDUOS CON DISTRIBUCIÓN NORMAL¶
SignificanceResult(statistic=417.01556109393744, pvalue=2.793968301284977e-91)
(0.11546272377541411, 0.0009999999999998899)

Los residuos no siguen una distribución normal.

75.30% de los residuos están dentro de ±1σ (esperado ≈ 68%)
93.94% de los residuos están dentro de ±2σ (esperado ≈ 95%)
98.79% de los residuos están dentro de ±3σ (esperado ≈ 99.7%)

Pero son bastante parecidos.

GRÁFICO DE RESIDUOS¶
No description has been provided for this image

SEGUNDO MODELO¶

SARIMAX Results
Dep. Variable: y No. Observations: 660
Model: SARIMAX(13, 0, 0)x(0, 1, [1, 2, 3, 4], 12) Log Likelihood -665.247
Date: Wed, 30 Apr 2025 AIC 1366.493
Time: 20:07:19 BIC 1447.023
Sample: 0 HQIC 1397.733
- 660
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 0.1420 0.035 4.086 0.000 0.074 0.210
ar.L2 -0.0189 0.037 -0.516 0.606 -0.090 0.053
ar.L3 0.0496 0.037 1.349 0.177 -0.022 0.122
ar.L4 -0.0694 0.042 -1.646 0.100 -0.152 0.013
ar.L5 -0.1131 0.050 -2.274 0.023 -0.211 -0.016
ar.L6 -0.1124 0.060 -1.872 0.061 -0.230 0.005
ar.L7 -0.0152 0.051 -0.299 0.765 -0.115 0.085
ar.L8 -0.0505 0.044 -1.154 0.249 -0.136 0.035
ar.L9 0.0363 0.035 1.030 0.303 -0.033 0.105
ar.L10 -0.0074 0.033 -0.225 0.822 -0.072 0.057
ar.L11 0.0991 0.037 2.677 0.007 0.027 0.172
ar.L12 0.5353 0.082 6.554 0.000 0.375 0.695
ar.L13 -0.0519 0.042 -1.247 0.213 -0.133 0.030
ma.S.L12 -1.3733 0.097 -14.094 0.000 -1.564 -1.182
ma.S.L24 0.3639 0.086 4.217 0.000 0.195 0.533
ma.S.L36 0.0957 0.060 1.599 0.110 -0.022 0.213
ma.S.L48 -0.0740 0.039 -1.888 0.059 -0.151 0.003
sigma2 0.4357 0.028 15.586 0.000 0.381 0.491
Ljung-Box (L1) (Q): 0.11 Jarque-Bera (JB): 344.46
Prob(Q): 0.74 Prob(JB): 0.00
Heteroskedasticity (H): 0.83 Skew: 0.87
Prob(H) (two-sided): 0.18 Kurtosis: 6.12


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

VERIFICACIÓN DE SUPUESTOS¶

PRINCIPIO DE PARSIMONIA¶
Hay coeficientes no significativos, no se cumple el principio de parsimonia
MODELO ADMISIBLE¶
El polinomio de la parte autoregresiva es: {'p1': -0.13691650540076578, 'p2': 0.01901377784903687, 'p3': -0.0500667165465603, 'p4': 0.069698229094628, 'p5': 0.11171395157927178, 'p6': 0.11009957522513744, 'p7': 0.010425775539985482, 'p8': 0.051137304909435626, 'p9': -0.03308758191153757, 'p10': 0.009267663653695928, 'p11': -0.1033592404899208, 'p12': -0.5385061478654415, 'p13': 0.046023546700947306}

El grado del polinomio es: 13

Raíces del polinomio característico: [11.88851581+0.j         -1.09964507+0.j         -0.92538665+0.52179453j
 -0.92538665-0.52179453j -0.52533631+0.91031656j -0.52533631-0.91031656j
 -0.02218345+1.04735764j -0.02218345-1.04735764j  0.5303036 +0.92193348j
  0.5303036 -0.92193348j  1.06751362+0.j          0.86474363+0.50810694j
  0.86474363-0.50810694j]

Módulo de las raíces: [11.88851581  1.09964507  1.06236057  1.06236057  1.05102544  1.05102544
  1.04759254  1.04759254  1.06357099  1.06357099  1.06751362  1.00297268
  1.00297268]

¿Las raíces están fuera del círculo unitario?  True

El modelo es estacionario

:)
El polinomio de la parte media móvil es: {'p12': -1.3756005193776482, 'p24': 0.3674412910433979, 'p36': 0.09734997881875754, 'p48': -0.07835471066657075}

El grado del polinomio es: 48

Raíces del polinomio característico: [-1.05628158e+00+0.2830298j  -1.05628158e+00-0.2830298j
 -1.06074970e+00+0.06258621j -1.06074970e+00-0.06258621j
 -1.00135998e+00+0.j         -7.73251787e-01+0.77325179j
 -7.73251787e-01-0.77325179j -9.49929287e-01+0.4761736j
 -9.49929287e-01-0.4761736j  -8.87343081e-01+0.58457609j
 -8.87343081e-01-0.58457609j -8.67203179e-01+0.50067999j
 -8.67203179e-01-0.50067999j -5.84576092e-01+0.88734308j
 -5.84576092e-01-0.88734308j -4.76173604e-01+0.94992929j
 -4.76173604e-01-0.94992929j -5.00679989e-01+0.86720318j
 -5.00679989e-01-0.86720318j -2.83029797e-01+1.05628158j
 -2.83029797e-01-1.05628158j -6.25862060e-02+1.0607497j
 -6.25862060e-02-1.0607497j   6.25862060e-02+1.0607497j
  6.25862060e-02-1.0607497j   2.16421979e-16+1.00135998j
  2.16421979e-16-1.00135998j  2.83029797e-01+1.05628158j
  2.83029797e-01-1.05628158j  4.76173604e-01+0.94992929j
  4.76173604e-01-0.94992929j  5.84576092e-01+0.88734308j
  5.84576092e-01-0.88734308j  5.00679989e-01+0.86720318j
  5.00679989e-01-0.86720318j  7.73251787e-01+0.77325179j
  7.73251787e-01-0.77325179j  1.05628158e+00+0.2830298j
  1.05628158e+00-0.2830298j   1.06074970e+00+0.06258621j
  1.06074970e+00-0.06258621j  1.00135998e+00+0.j
  8.87343081e-01+0.58457609j  8.87343081e-01-0.58457609j
  9.49929287e-01+0.4761736j   9.49929287e-01-0.4761736j
  8.67203179e-01+0.50067999j  8.67203179e-01-0.50067999j]

Módulo de las raíces: [1.09354316 1.09354316 1.06259444 1.06259444 1.00135998 1.09354316
 1.09354316 1.06259444 1.06259444 1.06259444 1.06259444 1.00135998
 1.00135998 1.06259444 1.06259444 1.06259444 1.06259444 1.00135998
 1.00135998 1.09354316 1.09354316 1.06259444 1.06259444 1.06259444
 1.06259444 1.00135998 1.00135998 1.09354316 1.09354316 1.06259444
 1.06259444 1.06259444 1.06259444 1.00135998 1.00135998 1.09354316
 1.09354316 1.09354316 1.09354316 1.06259444 1.06259444 1.00135998
 1.06259444 1.06259444 1.06259444 1.06259444 1.00135998 1.00135998]

¿Las raíces están fuera del círculo unitario?  True

El modelo es invertible

:)
RESIDUOS INDEPENDIENTES¶
lb_stat lb_pvalue
1 0.169555 NaN
2 0.204491 NaN
3 0.207502 NaN
4 0.727124 NaN
5 2.068440 NaN
6 2.452464 NaN
7 2.514097 NaN
8 2.809591 NaN
9 3.353825 NaN
10 4.344480 NaN
11 4.491955 NaN
12 4.756059 NaN
13 5.816217 NaN
14 6.013503 NaN
15 6.142261 NaN
16 6.454918 NaN
17 6.774386 NaN
18 6.783618 0.009200
19 6.786944 0.033592
20 6.809256 0.078232
21 7.196463 0.125863
22 7.428032 0.190703
23 7.441854 0.281909
24 7.447962 0.383773
25 7.506681 0.483077
26 8.263074 0.507868
27 8.898238 0.541788
28 9.010612 0.620913
29 9.533962 0.656772
30 9.544856 0.730662
31 9.545139 0.794626
32 11.177851 0.739888
33 11.194537 0.797321
34 13.243411 0.719745
35 13.251323 0.776431
36 13.698894 0.800953
37 13.804935 0.840247
38 14.263347 0.858024
39 14.567144 0.880135
40 15.189601 0.887710
41 16.919636 0.852069
42 16.949758 0.883608
43 16.977019 0.909776
44 17.083844 0.929034
45 17.970345 0.926894
46 18.039168 0.943470
47 18.078955 0.957241
48 18.964369 0.955485

Hay dependencia en los primeros dos residuos.

RESIDUOS CON MEDIA CERO¶
-0.04384260119451572
TtestResult(statistic=-1.6518512196908124, pvalue=0.09904125403907901, df=659)

La media de los residuos puede ser 0.

RESIDUOS CON VARIANZA CONSTANTE¶
(2.2724202057175913,
 0.1316941016400431,
 2.273361405689974,
 0.13209411797812018)

Los residuos tienen varianza constante.

RESIDUOS CON DISTRIBUCIÓN NORMAL¶
SignificanceResult(statistic=380.3670597932282, pvalue=2.537128226453911e-83)
(0.11315000343530657, 0.0009999999999998899)

Los residuos no siguen una distribución normal.

75.15% de los residuos están dentro de ±1σ (esperado ≈ 68%)
93.79% de los residuos están dentro de ±2σ (esperado ≈ 95%)
98.94% de los residuos están dentro de ±3σ (esperado ≈ 99.7%)

Pero son bastante parecidos.

GRÁFICO DE RESIDUOS¶
No description has been provided for this image

TERCER MODELO¶

SARIMAX Results
Dep. Variable: y No. Observations: 660
Model: SARIMAX(12, 0, 0)x(0, 1, [1, 2], 12) Log Likelihood -668.499
Date: Sun, 27 Apr 2025 AIC 1366.998
Time: 04:49:32 BIC 1434.107
Sample: 0 HQIC 1393.032
- 660
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 0.1241 0.033 3.797 0.000 0.060 0.188
ar.L2 -0.0278 0.037 -0.746 0.455 -0.101 0.045
ar.L3 0.0593 0.036 1.629 0.103 -0.012 0.131
ar.L4 -0.0777 0.043 -1.819 0.069 -0.161 0.006
ar.L5 -0.1077 0.050 -2.137 0.033 -0.206 -0.009
ar.L6 -0.1154 0.060 -1.914 0.056 -0.234 0.003
ar.L7 -0.0107 0.050 -0.214 0.831 -0.108 0.087
ar.L8 -0.0498 0.044 -1.129 0.259 -0.136 0.037
ar.L9 0.0450 0.036 1.263 0.207 -0.025 0.115
ar.L10 -0.0087 0.033 -0.263 0.793 -0.074 0.056
ar.L11 0.1075 0.035 3.042 0.002 0.038 0.177
ar.L12 0.5090 0.073 6.931 0.000 0.365 0.653
ma.S.L12 -1.3374 0.082 -16.355 0.000 -1.498 -1.177
ma.S.L24 0.3550 0.075 4.760 0.000 0.209 0.501
sigma2 0.4394 0.022 20.157 0.000 0.397 0.482
Ljung-Box (L1) (Q): 0.11 Jarque-Bera (JB): 385.54
Prob(Q): 0.74 Prob(JB): 0.00
Heteroskedasticity (H): 0.82 Skew: 0.91
Prob(H) (two-sided): 0.15 Kurtosis: 6.31


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

VERIFICACIÓN DE SUPUESTOS¶

PRINCIPIO DE PARSIMONIA¶
Hay coeficientes no significativos, no se cumple el principio de parsimonia
MODELO ADMISIBLE¶
El polinomio de la parte autoregresiva es: {'p1': -0.12409609720857327, 'p2': 0.02784753160299208, 'p3': -0.05931370060464292, 'p4': 0.07773595850485018, 'p5': 0.10769845417883445, 'p6': 0.1153846887487895, 'p7': 0.010651434003279378, 'p8': 0.04983749921571501, 'p9': -0.04496683215549213, 'p10': 0.008718941011892536, 'p11': -0.10747007955137028, 'p12': -0.5090305660410539}

El grado del polinomio es: 12

Raíces del polinomio característico: [-1.12406749+0.j         -0.93496549+0.52952748j -0.93496549-0.52952748j
 -0.52508333+0.91815394j -0.52508333-0.91815394j -0.01640444+1.05341344j
 -0.01640444-1.05341344j  0.53677332+0.92317355j  0.53677332-0.92317355j
  1.06348079+0.j          0.8644098 +0.50791775j  0.8644098 -0.50791775j]

Módulo de las raíces: [1.12406749 1.07450445 1.07450445 1.05769521 1.05769521 1.05354116
 1.05354116 1.06788342 1.06788342 1.06348079 1.00258902 1.00258902]

¿Las raíces están fuera del círculo unitario?  True

El modelo es estacionario

:)
El polinomio de la parte media móvil es: {'p12': -1.3373896067612656, 'p24': 0.3549819568174022}

El grado del polinomio es: 24

Raíces del polinomio característico: [-1.08759159e+00+0.j         -1.00234440e+00+0.j
 -9.41881945e-01+0.54379579j -9.41881945e-01-0.54379579j
 -8.68055716e-01+0.5011722j  -8.68055716e-01-0.5011722j
 -5.43795795e-01+0.94188195j -5.43795795e-01-0.94188195j
 -5.01172202e-01+0.86805572j -5.01172202e-01-0.86805572j
 -7.34340403e-16+1.08759159j -7.34340403e-16-1.08759159j
  9.15927823e-17+1.0023444j   9.15927823e-17-1.0023444j
  5.43795795e-01+0.94188195j  5.43795795e-01-0.94188195j
  5.01172202e-01+0.86805572j  5.01172202e-01-0.86805572j
  9.41881945e-01+0.54379579j  9.41881945e-01-0.54379579j
  1.08759159e+00+0.j          1.00234440e+00+0.j
  8.68055716e-01+0.5011722j   8.68055716e-01-0.5011722j ]

Módulo de las raíces: [1.08759159 1.0023444  1.08759159 1.08759159 1.0023444  1.0023444
 1.08759159 1.08759159 1.0023444  1.0023444  1.08759159 1.08759159
 1.0023444  1.0023444  1.08759159 1.08759159 1.0023444  1.0023444
 1.08759159 1.08759159 1.08759159 1.0023444  1.0023444  1.0023444 ]

¿Las raíces están fuera del círculo unitario?  True

El modelo es invertible

:)
RESIDUOS INDEPENDIENTES¶
lb_stat lb_pvalue
1 0.852590 NaN
2 1.026959 NaN
3 1.042580 NaN
4 1.330274 NaN
5 2.702490 NaN
6 2.993605 NaN
7 3.102314 NaN
8 3.441180 NaN
9 3.616131 NaN
10 4.475840 NaN
11 4.662731 NaN
12 4.990946 NaN
13 7.916099 NaN
14 8.223545 NaN
15 8.469831 0.003611
16 8.751457 0.012579
17 9.104454 0.027934
18 9.135493 0.057801
19 9.135514 0.103780
20 9.139624 0.165877
21 9.745116 0.203483
22 10.168911 0.253370
23 10.181542 0.335989
24 10.580242 0.391141
25 10.587480 0.478441
26 11.470408 0.489092
27 12.051651 0.523413
28 12.201265 0.590143
29 12.738996 0.622450
30 12.744188 0.691362
31 12.749628 0.752779
32 14.953637 0.665148
33 14.972683 0.724332
34 17.346330 0.630384
35 17.349312 0.689726
36 18.361216 0.684370
37 18.361346 0.737605
38 18.844803 0.760215
39 19.320376 0.781506
40 19.893036 0.796598
41 21.815050 0.746715
42 21.912901 0.785308
43 21.985648 0.820783
44 22.126466 0.849404
45 23.141840 0.843941
46 23.250141 0.870261
47 23.337109 0.893652
48 23.478694 0.912247

Hay dependencia en los primeros dos residuos.

RESIDUOS CON MEDIA CERO¶
-0.03975299918077584
TtestResult(statistic=-1.4907248613412258, pvalue=0.1365122225435534, df=659)

La media de los residuos puede ser 0.

RESIDUOS CON VARIANZA CONSTANTE¶
(2.3477675031968714,
 0.12546257027135715,
 2.3490090062319307,
 0.1258433113034253)

Los residuos tienen varianza constante.

RESIDUOS CON DISTRIBUCIÓN NORMAL¶
SignificanceResult(statistic=421.16509310160836, pvalue=3.508827740565011e-92)
(0.10731720679113188, 0.0009999999999998899)

Los residuos no siguen una distribución normal.

75.15% de los residuos están dentro de ±1σ (esperado ≈ 68%)
94.09% de los residuos están dentro de ±2σ (esperado ≈ 95%)
98.79% de los residuos están dentro de ±3σ (esperado ≈ 99.7%)

Pero son bastante parecidos.

GRÁFICO DE RESIDUOS¶
No description has been provided for this image

OTROS MODELOS QUE SE CREYÓ QUE PODÍAN SER¶

Luego de ver la FAC y FACP:

Valores de autocorrelacion significativos:
r1: 0.08495427799538309
r5: -0.09455404757011152
r12: -0.41418332950296094
r69: 0.13811523521698316
r73: -0.11256100076387932
r82: -0.09936326169869977
r139: 0.10081286249593209
r144: -0.103722152043645
Valores de autocorrelacion parcial significativos:
rho 1: 0.08508558290727704
rho 5: -0.08389897753926691
rho 12: -0.43583220424270375
rho 17: -0.12023508334631484
rho 24: -0.30097443805877616
rho 29: -0.10198920198664936
rho 36: -0.15950168604080825
rho 48: -0.14036369491687153
rho 57: -0.09168717760514207
rho 60: -0.1239606605892366
rho 61: 0.13800141862943802
rho 69: 0.10269095820413282
rho 72: -0.09462958643919699
rho 82: -0.08002594037040578
rho 85: -0.08198325512480845
rho 93: 0.092203788733838
rho 109: 0.09582379018763891
rho 113: -0.09105599732819755
rho 119: 0.09171771966671013
rho 120: -0.13558724808031059

Se pensó que podía ser este modelo, pero es muy malo.

SARIMAX Results
Dep. Variable: y No. Observations: 660
Model: SARIMAX([1, 5], 0, [1, 5])x(1, 1, 2, 12) Log Likelihood -5.099
Date: Sun, 27 Apr 2025 AIC 26.198
Time: 04:51:06 BIC 61.990
Sample: 0 HQIC 40.083
- 660
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 1.1476 32.468 0.035 0.972 -62.489 64.784
ar.L5 -0.8775 38.486 -0.023 0.982 -76.308 74.553
ma.L1 -0.9317 23.880 -0.039 0.969 -47.736 45.872
ma.L5 0.3314 36.742 0.009 0.993 -71.681 72.344
ar.S.L12 -0.3697 133.649 -0.003 0.998 -262.317 261.577
ma.S.L12 -0.1630 41.007 -0.004 0.997 -80.536 80.210
ma.S.L24 -0.7558 42.337 -0.018 0.986 -83.735 82.223
sigma2 0.4434 8.359 0.053 0.958 -15.941 16.827
Ljung-Box (L1) (Q): 294.50 Jarque-Bera (JB): 2134861.83
Prob(Q): 0.00 Prob(JB): 0.00
Heteroskedasticity (H): 0.00 Skew: 15.72
Prob(H) (two-sided): 0.00 Kurtosis: 282.43


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
[2] Covariance matrix is singular or near-singular, with condition number 6.39e+17. Standard errors may be unstable.
-7.786026841214317e+48

También este, pero no es invertible

SARIMAX Results
Dep. Variable: y No. Observations: 660
Model: SARIMAX(0, 0, [5])x(1, 1, [1, 2], 12) Log Likelihood -708.534
Date: Sun, 27 Apr 2025 AIC 1427.068
Time: 04:51:13 BIC 1449.437
Sample: 0 HQIC 1435.745
- 660
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ma.L5 -0.1707 0.044 -3.861 0.000 -0.257 -0.084
ar.S.L12 -0.8858 0.147 -6.011 0.000 -1.175 -0.597
ma.S.L12 0.2236 0.140 1.599 0.110 -0.050 0.498
ma.S.L24 -0.6343 0.088 -7.209 0.000 -0.807 -0.462
sigma2 0.5151 0.019 27.358 0.000 0.478 0.552
Ljung-Box (L1) (Q): 10.39 Jarque-Bera (JB): 283.30
Prob(Q): 0.00 Prob(JB): 0.00
Heteroskedasticity (H): 0.83 Skew: 0.66
Prob(H) (two-sided): 0.18 Kurtosis: 5.96


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
El polinomio de la parte media móvil es: {'p5': -0.17066687867489047, 'p12': 0.2235561375559531, 'p24': -0.6342522946355272}

El grado del polinomio es: 24

Raíces del polinomio característico: [-1.03809773+0.j         -0.97339645+0.26906311j -0.97339645-0.26906311j
 -0.88525875+0.51576505j -0.88525875-0.51576505j -0.71259807+0.7035491j
 -0.71259807-0.7035491j  -0.52270681+0.89327539j -0.52270681-0.89327539j
 -0.26081491+0.98110341j -0.26081491-0.98110341j  0.00733524+1.03148712j
  0.00733524-1.03148712j  1.02333621+0.j          0.97341295+0.25216589j
  0.97341295-0.25216589j  0.89999742+0.51572558j  0.89999742-0.51572558j
  0.71256537+0.7204749j   0.71256537-0.7204749j   0.50801366+0.89331475j
  0.50801366-0.89331475j  0.2608311 +0.96414954j  0.2608311 -0.96414954j]

Módulo de las raíces: [1.03809773 1.0098988  1.0098988  1.02454704 1.02454704 1.00138771
 1.00138771 1.0349702  1.0349702  1.01517895 1.01517895 1.03151321
 1.03151321 1.02333621 1.00554483 1.00554483 1.03728888 1.03728888
 1.01332793 1.01332793 1.02766197 1.02766197 0.99880789 0.99880789]

¿Las raíces están fuera del círculo unitario?  False

El modelo no es invertible

también este, pero no es invertible.

SARIMAX Results
Dep. Variable: y No. Observations: 660
Model: SARIMAX([12], 0, [5])x(0, 1, [1, 2], 12) Log Likelihood -708.534
Date: Sun, 27 Apr 2025 AIC 1427.068
Time: 04:51:22 BIC 1449.437
Sample: 0 HQIC 1435.745
- 660
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L12 -0.8858 0.147 -6.011 0.000 -1.175 -0.597
ma.L5 -0.1707 0.044 -3.861 0.000 -0.257 -0.084
ma.S.L12 0.2236 0.140 1.599 0.110 -0.050 0.498
ma.S.L24 -0.6343 0.088 -7.208 0.000 -0.807 -0.462
sigma2 0.5151 0.019 27.358 0.000 0.478 0.552
Ljung-Box (L1) (Q): 10.39 Jarque-Bera (JB): 283.30
Prob(Q): 0.00 Prob(JB): 0.00
Heteroskedasticity (H): 0.83 Skew: 0.66
Prob(H) (two-sided): 0.18 Kurtosis: 5.96


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).
El polinomio de la parte media móvil es: {'p5': -0.1706643002131486, 'p12': 0.22358740803256344, 'p24': -0.6342671085676491}

El grado del polinomio es: 24

Raíces del polinomio característico: [-1.03809804+0.j         -0.97339404+0.26906235j -0.97339404-0.26906235j
 -0.88525946+0.5157653j  -0.88525946-0.5157653j  -0.71259631+0.70354748j
 -0.71259631-0.70354748j -0.52270685+0.89327583j -0.52270685-0.89327583j
 -0.26081426+0.98110091j -0.26081426-0.98110091j  0.00733501+1.03148763j
  0.00733501-1.03148763j  1.023337  +0.j          0.97341054+0.25216538j
  0.97341054-0.25216538j  0.89999765+0.51572583j  0.89999765-0.51572583j
  0.7125636 +0.72047304j  0.7125636 -0.72047304j  0.50801417+0.89331519j
  0.50801417-0.89331519j  0.26083046+0.96414729j  0.26083046-0.96414729j]

Módulo de las raíces: [1.03809804 1.00989628 1.00989628 1.02454778 1.02454778 1.00138532
 1.00138532 1.0349706  1.0349706  1.01517637 1.01517637 1.03151371
 1.03151371 1.023337   1.00554237 1.00554237 1.03728921 1.03728921
 1.01332536 1.01332536 1.0276626  1.0276626  0.99880555 0.99880555]

¿Las raíces están fuera del círculo unitario?  False

El modelo no es invertible

MODELO PARA LA SERIE TRANSFORMADA¶

Valores de autocorrelacion significativos:
r1: 0.1881206416574384
r2: 0.12601374062799692
r6: -0.10023577930193152
r12: -0.40245032162240263
r33: 0.09802046981427195
r45: -0.11985655705785987
Valores de autocorrelacion parcial significativos:
rho 1: 0.18841139999075743
rho 2: 0.09425081653205467
rho 6: -0.0848567418000625
rho 12: -0.4319892618507997
rho 13: 0.08136260875048439
rho 17: -0.11385209887045368
rho 22: -0.08401445717662205
rho 24: -0.2809725408252081
rho 25: 0.10755279636113212
rho 30: -0.09673228306484968
rho 33: 0.1105360450418002
rho 36: -0.14602967562386657
rho 42: -0.11769525648448687
rho 46: -0.09378255882994088
rho 48: -0.16710968828081296
rho 50: 0.09151867173223789
rho 55: 0.09278816335947565
rho 60: -0.09971581325023068
rho 72: -0.13002571917786468
rho 78: -0.07964448430687596
rho 84: -0.07986957194068202
rho 92: -0.13966165907823416
rho 93: 0.11600081821072342
rho 94: -0.1221857229746919
rho 113: -0.08611614522897519
rho 118: -0.08199830520693245
rho 120: -0.11627799482364788

PRIMER MODELO¶

Por la FAC y FACP se cree que puede ser el siguiente modelo:

SARIMAX Results
Dep. Variable: y No. Observations: 660
Model: SARIMAX([1, 2, 6], 0, 1)x(1, 1, [1, 2, 3, 4, 5, 6], 12) Log Likelihood -617.741
Date: Sun, 27 Apr 2025 AIC 1259.482
Time: 20:29:33 BIC 1313.169
Sample: 0 HQIC 1280.309
- 660
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 0.3566 0.144 2.482 0.013 0.075 0.638
ar.L2 0.0211 0.062 0.342 0.732 -0.100 0.142
ar.L6 -0.2754 0.036 -7.653 0.000 -0.346 -0.205
ma.L1 -0.1092 0.149 -0.731 0.465 -0.402 0.184
ar.S.L12 -0.8536 0.246 -3.468 0.001 -1.336 -0.371
ma.S.L12 0.1009 0.245 0.412 0.680 -0.379 0.581
ma.S.L24 -0.7034 0.191 -3.691 0.000 -1.077 -0.330
ma.S.L36 0.0214 0.049 0.435 0.664 -0.075 0.118
ma.S.L48 0.0154 0.054 0.286 0.775 -0.090 0.121
ma.S.L60 0.0169 0.053 0.319 0.750 -0.087 0.121
ma.S.L72 0.0063 0.055 0.114 0.909 -0.101 0.114
sigma2 0.3874 0.020 19.120 0.000 0.348 0.427
Ljung-Box (L1) (Q): 0.02 Jarque-Bera (JB): 10.86
Prob(Q): 0.88 Prob(JB): 0.00
Heteroskedasticity (H): 0.83 Skew: -0.20
Prob(H) (two-sided): 0.17 Kurtosis: 3.50


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

VERIFICACIÓN DE SUPUESTOS¶

RESIDUOS INDEPENDIENTES¶
lb_stat lb_pvalue
1 0.066845 NaN
2 1.537653 NaN
3 1.886894 NaN
4 4.744317 NaN
5 19.994274 NaN
6 20.002045 NaN
7 20.551066 NaN
8 20.563646 NaN
9 25.131429 NaN
10 25.236634 NaN
11 31.550176 NaN
12 31.652621 1.843656e-08
13 31.779184 1.256718e-07
14 31.817201 5.718884e-07
15 31.850384 2.052629e-06
16 34.932704 1.551903e-06
17 42.942269 1.197523e-07
18 48.759895 2.528439e-08
19 50.340543 3.514875e-08
20 50.340543 9.294961e-08
21 51.913020 1.184210e-07
22 52.744140 1.998096e-07
23 57.847442 5.554601e-08
24 57.847451 1.270340e-07
25 59.668596 1.341066e-07
26 59.677107 2.865712e-07
27 59.912799 5.413768e-07
28 60.290463 9.410050e-07
29 60.475407 1.713942e-06
30 69.299462 1.202209e-07
31 69.362452 2.313866e-07
32 69.561924 4.130052e-07
33 73.056696 2.166748e-07
34 74.446295 2.456685e-07
35 76.961663 1.840763e-07
36 76.961687 3.388905e-07
37 78.817354 3.185844e-07
38 80.532525 3.135244e-07
39 83.152029 2.243965e-07
40 83.267861 3.812489e-07
41 84.188435 4.853694e-07
42 86.097859 4.396425e-07
43 86.761902 6.023649e-07
44 86.884845 9.763344e-07
45 90.327760 5.298933e-07
46 91.250891 6.557028e-07
47 98.721616 9.467501e-08
48 98.935868 1.493370e-07

Hay dependencia en los residuos.

RESIDUOS CON MEDIA CERO¶
-0.017262744929477162
TtestResult(statistic=-0.6945529062973945, pvalue=0.4875803217531771, df=659)

La media de los residuos puede ser 0.

GRÁFICO DE RESIDUOS¶
No description has been provided for this image

SEGUNDO MODELO¶

Pero no captura completamente la dependencia temporal, por lo que se agregó la parte completa para la parte AR

SARIMAX Results
Dep. Variable: y No. Observations: 660
Model: SARIMAX(6, 0, 1)x(1, 1, 1, 12) Log Likelihood -602.344
Date: Sun, 27 Apr 2025 AIC 1224.688
Time: 20:29:37 BIC 1269.426
Sample: 0 HQIC 1242.043
- 660
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 0.4302 0.148 2.908 0.004 0.140 0.720
ar.L2 0.0474 0.067 0.702 0.483 -0.085 0.180
ar.L3 -0.0470 0.047 -0.995 0.320 -0.140 0.046
ar.L4 -0.0870 0.043 -2.013 0.044 -0.172 -0.002
ar.L5 -0.1800 0.045 -3.968 0.000 -0.269 -0.091
ar.L6 -0.2197 0.056 -3.904 0.000 -0.330 -0.109
ma.L1 -0.1680 0.154 -1.089 0.276 -0.470 0.134
ar.S.L12 0.0208 0.046 0.451 0.652 -0.070 0.111
ma.S.L12 -0.8907 0.029 -30.470 0.000 -0.948 -0.833
sigma2 0.3657 0.019 19.149 0.000 0.328 0.403
Ljung-Box (L1) (Q): 0.02 Jarque-Bera (JB): 5.80
Prob(Q): 0.89 Prob(JB): 0.05
Heteroskedasticity (H): 0.81 Skew: -0.17
Prob(H) (two-sided): 0.12 Kurtosis: 3.32


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

VERIFICACIÓN DE SUPUESTOS¶

RESIDUOS INDEPENDIENTES¶
lb_stat lb_pvalue
1 0.078029 NaN
2 0.081745 NaN
3 0.110251 NaN
4 0.667736 NaN
5 1.809556 NaN
6 3.209615 NaN
7 3.289842 NaN
8 3.816187 NaN
9 6.473299 NaN
10 7.880155 0.004998
11 9.034590 0.010919
12 9.198592 0.026764
13 9.646686 0.046819
14 9.995740 0.075356
15 10.036647 0.123117
16 10.813321 0.146972
17 20.147979 0.009790
18 24.051508 0.004221
19 24.928482 0.005483
20 25.176289 0.008590
21 25.707594 0.011804
22 30.300176 0.004262
23 32.585774 0.003304
24 32.718036 0.005134
25 33.218651 0.006912
26 33.280245 0.010384
27 33.486398 0.014565
28 33.781554 0.019498
29 33.785406 0.027614
30 39.683798 0.008123
31 39.694743 0.011734
32 39.795172 0.016202
33 43.588079 0.008516
34 43.726951 0.011633
35 45.030235 0.011682
36 45.950944 0.012878
37 49.181595 0.007969
38 50.128815 0.008747
39 52.006371 0.007605
40 52.303949 0.009733
41 53.661655 0.009590
42 55.762429 0.007912
43 56.156343 0.009780
44 56.305459 0.012669
45 60.129401 0.007053
46 63.211636 0.004624
47 69.223074 0.001459
48 69.868377 0.001735

Y el modelo captura correctamente la dependencia temporal.

El polinomio de la parte autoregresiva es: {'p1': -0.4302180830507938, 'p2': -0.04735397876460526, 'p3': 0.04698747454292404, 'p4': 0.08700716618181399, 'p5': 0.1799715794603173, 'p6': 0.21973455421960683, 'p12': -0.02082020047854175}

Raíces del polinomio característico: [-1.51585027+0.j         -1.16433253+0.66675602j -1.16433253-0.66675602j
 -0.70531602+1.36923083j -0.70531602-1.36923083j  0.00589534+1.27354789j
  0.00589534-1.27354789j  0.89804478+1.3208691j   0.89804478-1.3208691j
  1.62535851+0.j          0.91095431+0.5227982j   0.91095431-0.5227982j ]

Módulo de las raíces: [1.51585027 1.34172793 1.34172793 1.54021549 1.54021549 1.27356154
 1.27356154 1.59724125 1.59724125 1.62535851 1.0503122  1.0503122 ]

¿Las raíces están fuera del círculo unitario?  True

El modelo es estacionario

:)
El polinomio de la parte media móvil es: {'p1': -0.16798341167047423, 'p12': -0.8906607240754797}

Raíces del polinomio característico: [-1.02312938+0.j         -0.88152548+0.51636208j -0.88152548-0.51636208j
 -0.49857917+0.88672189j -0.49857917-0.88672189j  0.01419601+1.01059688j
  0.01419601-1.01059688j  0.51292531+0.86213479j  0.51292531-0.86213479j
  0.99443323+0.j          0.86733141+0.49151162j  0.86733141-0.49151162j]

Módulo de las raíces: [1.02312938 1.02162467 1.02162467 1.01727917 1.01727917 1.01069658
 1.01069658 1.00317933 1.00317933 0.99443323 0.99691898 0.99691898]

¿Las raíces están fuera del círculo unitario?  False

El modelo no es invertible

TERCER MODELO¶

SARIMAX Results
Dep. Variable: y No. Observations: 660
Model: SARIMAX(6, 0, 0)x(1, 1, [1], 12) Log Likelihood -603.289
Date: Sun, 27 Apr 2025 AIC 1224.578
Time: 20:29:39 BIC 1264.843
Sample: 0 HQIC 1240.198
- 660
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 0.2721 0.034 7.929 0.000 0.205 0.339
ar.L2 0.1022 0.041 2.473 0.013 0.021 0.183
ar.L3 -0.0255 0.040 -0.644 0.520 -0.103 0.052
ar.L4 -0.0900 0.041 -2.215 0.027 -0.170 -0.010
ar.L5 -0.1974 0.039 -5.011 0.000 -0.275 -0.120
ar.L6 -0.2604 0.038 -6.796 0.000 -0.336 -0.185
ar.S.L12 0.0196 0.046 0.426 0.670 -0.071 0.110
ma.S.L12 -0.8851 0.029 -30.327 0.000 -0.942 -0.828
sigma2 0.3670 0.019 19.130 0.000 0.329 0.405
Ljung-Box (L1) (Q): 0.22 Jarque-Bera (JB): 6.43
Prob(Q): 0.64 Prob(JB): 0.04
Heteroskedasticity (H): 0.80 Skew: -0.19
Prob(H) (two-sided): 0.10 Kurtosis: 3.30


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

VERIFICACIÓN DE SUPUESTOS¶

PRINCIPIO DE PARSIMONIA¶
Hay coeficientes no significativos, no se cumple el principio de parsimonia
MODELO ADMISIBLE¶
El polinomio de la parte autoregresiva es: {'p1': -0.2721484850682655, 'p2': -0.10222956018637822, 'p3': 0.02552839449395747, 'p4': 0.09002860439631356, 'p5': 0.1974326132169234, 'p6': 0.2604147375399908, 'p12': -0.01958784345380709}

Raíces del polinomio característico: [-1.5324563 +0.j         -0.72632726+1.38599378j -0.72632726-1.38599378j
 -1.13728973+0.63130766j -1.13728973-0.63130766j  0.91165759+1.36955392j
  0.91165759-1.36955392j -0.03290209+1.25835037j -0.03290209-1.25835037j
  1.6720367 +0.j          0.91507128+0.53286918j  0.91507128-0.53286918j]

Módulo de las raíces: [1.5324563  1.56477795 1.56477795 1.30076027 1.30076027 1.64523479
 1.64523479 1.25878045 1.25878045 1.6720367  1.0589169  1.0589169 ]

¿Las raíces están fuera del círculo unitario?  True

El modelo es estacionario

:)
El polinomio de la parte media móvil es: {'p12': -0.8851100185522583}

Raíces del polinomio característico: [-1.01022217e+00+0.j         -8.74878063e-01+0.50511109j
 -8.74878063e-01-0.50511109j -5.05111085e-01+0.87487806j
 -5.05111085e-01-0.87487806j -5.55111512e-17+1.01022217j
 -5.55111512e-17-1.01022217j  5.05111085e-01+0.87487806j
  5.05111085e-01-0.87487806j  1.01022217e+00+0.j
  8.74878063e-01+0.50511109j  8.74878063e-01-0.50511109j]

Módulo de las raíces: [1.01022217 1.01022217 1.01022217 1.01022217 1.01022217 1.01022217
 1.01022217 1.01022217 1.01022217 1.01022217 1.01022217 1.01022217]

¿Las raíces están fuera del círculo unitario?  True

El modelo es invertible

:)
RESIDUOS INDEPENDIENTES¶
lb_stat lb_pvalue
1 0.001254 NaN
2 0.127263 NaN
3 0.145069 NaN
4 0.355249 NaN
5 1.452317 NaN
6 3.552256 NaN
7 3.845680 NaN
8 4.082125 NaN
9 6.692169 0.009684
10 7.832130 0.019919
11 9.102569 0.027958
12 9.256310 0.055003
13 9.544643 0.089215
14 9.734154 0.136303
15 9.887684 0.195028
16 10.818577 0.212192
17 19.823382 0.019034
18 23.899218 0.007872
19 24.703424 0.010073
20 25.013777 0.014758
21 25.631668 0.019043
22 29.998524 0.007635
23 32.696622 0.005169
24 32.798350 0.007858
25 33.458538 0.009854
26 33.506221 0.014484
27 33.689446 0.019989
28 33.907394 0.026758
29 33.908062 0.037069
30 40.197221 0.010252
31 40.244748 0.014426
32 40.383594 0.019444
33 44.092471 0.010590
34 44.309675 0.013997
35 45.634276 0.013925
36 46.439778 0.015687
37 49.474952 0.010282
38 50.724475 0.010416
39 52.415648 0.009474
40 52.583380 0.012372
41 53.835941 0.012450
42 55.965773 0.010224
43 56.386506 0.012439
44 56.652499 0.015534
45 60.570646 0.008572
46 63.146117 0.006367
47 69.344542 0.001975
48 69.962021 0.002346

Hay dependencia en los primeros dos residuos.

RESIDUOS CON MEDIA CERO¶
-0.03293748637459734
TtestResult(statistic=-1.3520655710406162, pvalue=0.1768181651558442, df=659)

La media de los residuos puede ser 0.

RESIDUOS CON VARIANZA CONSTANTE¶
(8.59265039527976,
 0.0033752241057990228,
 8.679613399395905,
 0.003331636731645561)

Los residuos tienen varianza constante.

RESIDUOS CON DISTRIBUCIÓN NORMAL¶
SignificanceResult(statistic=6.693009944604184, pvalue=0.03520718942157648)
(0.02689680139562356, 0.374813270764381)

Los residuos puede que sigan una distribución normal.

70.00% de los residuos están dentro de ±1σ (esperado ≈ 68%)
95.30% de los residuos están dentro de ±2σ (esperado ≈ 95%)
99.55% de los residuos están dentro de ±3σ (esperado ≈ 99.7%)

Pero son bastante parecidos.

GRÁFICO DE RESIDUOS¶
No description has been provided for this image

MODELOS SELECCIONADOS¶

MODELO SELECCIONADO OBTENIDO CON LA SERIE SIN TRANSFORMAR¶

Para la serie sin transformar, este es el modelo seleccionado:

$$\text{ARIMA}(13, 0, 0)(0, 1, 4)_{12}$$

Nos quedamos con este modelo porque es el que tiene los p-valores más altos en la prueba de independencia de los residuos. Cumple todos los supuestos, menos el principio de parsimonia, y sus residuos siguen una distribución normal relajada. Tiene un $\text{AIC} = 1367.369$

SARIMAX Results
Dep. Variable: y No. Observations: 660
Model: SARIMAX(13, 0, 0)x(0, 1, [1, 2, 3, 4], 12) Log Likelihood -665.247
Date: Sun, 27 Apr 2025 AIC 1366.493
Time: 20:30:25 BIC 1447.023
Sample: 0 HQIC 1397.733
- 660
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 0.1420 0.035 4.086 0.000 0.074 0.210
ar.L2 -0.0189 0.037 -0.516 0.606 -0.090 0.053
ar.L3 0.0496 0.037 1.349 0.177 -0.022 0.122
ar.L4 -0.0694 0.042 -1.646 0.100 -0.152 0.013
ar.L5 -0.1131 0.050 -2.274 0.023 -0.211 -0.016
ar.L6 -0.1124 0.060 -1.872 0.061 -0.230 0.005
ar.L7 -0.0152 0.051 -0.299 0.765 -0.115 0.085
ar.L8 -0.0505 0.044 -1.154 0.249 -0.136 0.035
ar.L9 0.0363 0.035 1.030 0.303 -0.033 0.105
ar.L10 -0.0074 0.033 -0.225 0.822 -0.072 0.057
ar.L11 0.0991 0.037 2.677 0.007 0.027 0.172
ar.L12 0.5353 0.082 6.554 0.000 0.375 0.695
ar.L13 -0.0519 0.042 -1.247 0.213 -0.133 0.030
ma.S.L12 -1.3733 0.097 -14.094 0.000 -1.564 -1.182
ma.S.L24 0.3639 0.086 4.217 0.000 0.195 0.533
ma.S.L36 0.0957 0.060 1.599 0.110 -0.022 0.213
ma.S.L48 -0.0740 0.039 -1.888 0.059 -0.151 0.003
sigma2 0.4357 0.028 15.586 0.000 0.381 0.491
Ljung-Box (L1) (Q): 0.11 Jarque-Bera (JB): 344.46
Prob(Q): 0.74 Prob(JB): 0.00
Heteroskedasticity (H): 0.83 Skew: 0.87
Prob(H) (two-sided): 0.18 Kurtosis: 6.12


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

MODELO SELECCIONADO OBTENIDO CON LA SERIE TRANSFORMADA¶

Para la serie transformada, este es el modelo seleccionado:

$$\text{ARIMA}(6, 0, 0)(1, 1, 1)_{12}$$

Cumple todos los supuestos, menos el principio de parsimonia, y tiene un $\text{AIC} = 1224.578$

SARIMAX Results
Dep. Variable: y No. Observations: 660
Model: SARIMAX(6, 0, 0)x(1, 1, [1], 12) Log Likelihood -603.289
Date: Sun, 27 Apr 2025 AIC 1224.578
Time: 20:30:27 BIC 1264.843
Sample: 0 HQIC 1240.198
- 660
Covariance Type: opg
coef std err z P>|z| [0.025 0.975]
ar.L1 0.2721 0.034 7.929 0.000 0.205 0.339
ar.L2 0.1022 0.041 2.473 0.013 0.021 0.183
ar.L3 -0.0255 0.040 -0.644 0.520 -0.103 0.052
ar.L4 -0.0900 0.041 -2.215 0.027 -0.170 -0.010
ar.L5 -0.1974 0.039 -5.011 0.000 -0.275 -0.120
ar.L6 -0.2604 0.038 -6.796 0.000 -0.336 -0.185
ar.S.L12 0.0196 0.046 0.426 0.670 -0.071 0.110
ma.S.L12 -0.8851 0.029 -30.327 0.000 -0.942 -0.828
sigma2 0.3670 0.019 19.130 0.000 0.329 0.405
Ljung-Box (L1) (Q): 0.22 Jarque-Bera (JB): 6.43
Prob(Q): 0.64 Prob(JB): 0.04
Heteroskedasticity (H): 0.80 Skew: -0.19
Prob(H) (two-sided): 0.10 Kurtosis: 3.30


Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

PRÓNOSTICO¶

MÉTODO FORECAST¶

FORECAST¶

INVERSA DE LA TRASNFORMACIÓN PARA EL PRONÓSTICO¶

mu: 784.0136363636364
sigma: 880.4446163963537

INTERVALOS DE CONFIANZA¶

INVERSA DE LA TRANSFORMACIÓN PARA EL IC¶

RESULTADO OBTENIDO CON SERIE SIN TRANSFORMAR¶
Pronóstico Límite Inferior Límite Superior Real Residuos
2008-12-31 540.502158 -601.447098 1682.451414 74 -466.502158
2009-01-31 201.870616 -951.512847 1355.254079 148 -53.870616
2009-02-28 218.349640 -935.034594 1371.733875 8 -210.349640
2009-03-31 666.919279 -487.716661 1821.555218 64 -602.919279
2009-04-30 824.911936 -331.463983 1981.287855 436 -388.911936
2009-05-31 1406.446553 240.377962 2572.515143 2627 1220.553447
2009-06-30 1256.178392 78.628305 2433.728478 972 -284.178392
2009-07-31 1262.636919 83.406848 2441.866990 1474 211.363081
2009-08-31 1259.408611 77.783466 2441.033756 3161 1901.591389
2009-09-30 736.647934 -445.287230 1918.583099 1190 453.352066
2009-10-31 307.230864 -875.186093 1489.647821 155 -152.230864
2009-11-30 340.113522 -851.828685 1532.055730 51 -289.113522
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RESULTADO CON LA SERIE TRANSFORMADA¶
Pronóstico Límite Inferior Límite Superior Real Residuos
2008-12-31 44.953714 -1.768799 634.580471 74 29.046286
2009-01-31 86.154521 -1.056604 965.287325 148 61.845479
2009-02-28 68.232307 -1.477922 872.355113 8 -60.232307
2009-03-31 131.127054 -0.297873 1260.040847 64 -67.127054
2009-04-30 428.426175 15.699218 2617.210481 436 7.573825
2009-05-31 763.114253 47.562078 3972.644947 2627 1863.885747
2009-06-30 779.087074 37.660483 4409.534391 972 192.912926
2009-07-31 869.768181 43.029260 4890.317654 1474 604.231819
2009-08-31 900.651856 44.624350 5062.589225 3161 2260.348144
2009-09-30 485.254867 10.547197 3399.343448 1190 704.745133
2009-10-31 155.744319 -0.807449 1736.882669 155 -0.744319
2009-11-30 108.943275 -1.512804 1460.964464 51 -57.943275
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COMPARACIÓN¶
Pronóstico Límite Inferior Límite Superior Pronóstico Transformado Límite Inferior Transformado Límite Superior Transformado Valores Reales Residuos Sin Transformar Residuos del Transformados
2008-12-31 540.502158 -601.447098 1682.451414 44.953714 -1.768799 634.580471 74 -466.502158 29.046286
2009-01-31 201.870616 -951.512847 1355.254079 86.154521 -1.056604 965.287325 148 -53.870616 61.845479
2009-02-28 218.349640 -935.034594 1371.733875 68.232307 -1.477922 872.355113 8 -210.349640 -60.232307
2009-03-31 666.919279 -487.716661 1821.555218 131.127054 -0.297873 1260.040847 64 -602.919279 -67.127054
2009-04-30 824.911936 -331.463983 1981.287855 428.426175 15.699218 2617.210481 436 -388.911936 7.573825
2009-05-31 1406.446553 240.377962 2572.515143 763.114253 47.562078 3972.644947 2627 1220.553447 1863.885747
2009-06-30 1256.178392 78.628305 2433.728478 779.087074 37.660483 4409.534391 972 -284.178392 192.912926
2009-07-31 1262.636919 83.406848 2441.866990 869.768181 43.029260 4890.317654 1474 211.363081 604.231819
2009-08-31 1259.408611 77.783466 2441.033756 900.651856 44.624350 5062.589225 3161 1901.591389 2260.348144
2009-09-30 736.647934 -445.287230 1918.583099 485.254867 10.547197 3399.343448 1190 453.352066 704.745133
2009-10-31 307.230864 -875.186093 1489.647821 155.744319 -0.807449 1736.882669 155 -152.230864 -0.744319
2009-11-30 340.113522 -851.828685 1532.055730 108.943275 -1.512804 1460.964464 51 -289.113522 -57.943275
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PRONÓSTICO ÓPTIMO¶

Se calcula en 04-Calculo-del-Pronostico-Optimo. Será calculado para el primer modelo, porque el segundo tiene parte autoregresiva estacional y no estacional, lo que hace que el polinomio sea muy grande.

$$ \text{ARIMA}(13,0,0) \times (0,1,4)_{12} \text{ para $X_t$} $$

con coeficientes:

  • $\phi_1 = -0.136917$
  • $\phi_2 = 0.019014$
  • $\phi_3 = -0.050067$
  • $\phi_4 = 0.069698$
  • $\phi_5 = 0.111714$
  • $\phi_6 = 0.110100$
  • $\phi_7 = 0.010426$
  • $\phi_8 = 0.051137$
  • $\phi_9 = -0.033088$
  • $\phi_{10} = 0.009268$
  • $\phi_{11} = -0.103359$
  • $\phi_{12} = -0.538506$
  • $\phi_{13} = 0.046024$

y para la parte estacional MA:

  • $\Theta_1 = 1.375601$
  • $\Theta_2 = -0.367441$
  • $\Theta_3 = -0.097350$
  • $\Theta_4 = 0.078355$
$$ \Phi_0(B^{12}) \, \phi_13(B) \, \nabla^0 \nabla_{12}^1 X_t = \Theta_4(B^{12}) \, \theta_0(B) \, \varepsilon_t, $$

Donde $\varepsilon_t \sim \mathcal{N}(0, 1)$

OBTENCIÓN DEL PRONÓSTICO ÓPTIMO¶

DESARROLLO PARA EL PRONÓSTICO ÓPTIMO¶

Del lado izquierdo:

$$ (1 - \phi_1 B - \phi_2 B^2 - \cdots - \phi_{13} B^{13})(X_t - X_{t-12}) = X_t - X_{t-12} - \phi_1 (X_{t-1} - X_{t-13}) - \phi_2 (X_{t-2} - X_{t-14}) - \cdots - \phi_{13}(X_{t-13} - X_{t-25}) $$$$ = X_t - X_{t-12} - \sum_{h=1}^{13} \phi_k (X_{t-h} - X_{t-h-12}) $$

Del lado derecho:

$$ (1 - \Theta_1 B^{12} - \Theta_2 B^{24} - \Theta_3 B^{36} - \Theta_4 B^{48})\varepsilon_t = \varepsilon_t - \Theta_1 \varepsilon_{t-12} - \Theta_2 \varepsilon_{t-24} - \Theta_3 \varepsilon_{t-36} - \Theta_4 \varepsilon_{t-48} $$

despejando $X_t$:

$$ X_t = X_{t-12} + \sum_{k=1}^{13} \phi_k (X_{t-k} - X_{t-k-12}) + \varepsilon_t - \Theta_1 \varepsilon_{t-12} - \Theta_2 \varepsilon_{t-24} - \Theta_3 \varepsilon_{t-36} - \Theta_4 \varepsilon_{t-48} $$

Por lo que, el pronóstico óptimo sería:

$$ X_t(h) = \underset{t}{\mathbb{E}}[X_{t+h-12}] + \sum_{k=1}^{13} \phi_k \left( \underset{t}{\mathbb{E}}[X_{t+h-k}] - \underset{t}{\mathbb{E}}[X_{t+h-k-12}] \right) + \underset{t}{\mathbb{E}}[\varepsilon_{t+h}] - \sum_{j=1}^4 \Theta_j \, \underset{t}{\mathbb{E}}[\varepsilon_{t+h-12j}] $$

El polinomio con las $\phi_k$ distribuidas:

$\phi_1 \underset{t}{\mathbb{E}}[X_{t+h-1}] - \phi_1 \underset{t}{\mathbb{E}}[X_{t+h-13}] +$ $\phi_2 \underset{t}{\mathbb{E}}[X_{t+h-2}] - \phi_2 \underset{t}{\mathbb{E}}[X_{t+h-14}] +$ $\phi_3 \underset{t}{\mathbb{E}}[X_{t+h-3}] - \phi_3 \underset{t}{\mathbb{E}}[X_{t+h-15}] +$ $\phi_4 \underset{t}{\mathbb{E}}[X_{t+h-4}] - \phi_4 \underset{t}{\mathbb{E}}[X_{t+h-16}] +$ $\phi_5 \underset{t}{\mathbb{E}}[X_{t+h-5}] - \phi_5 \underset{t}{\mathbb{E}}[X_{t+h-17}] +$ $\phi_6 \underset{t}{\mathbb{E}}[X_{t+h-6}] - \phi_6 \underset{t}{\mathbb{E}}[X_{t+h-18}] +$ $\phi_7 \underset{t}{\mathbb{E}}[X_{t+h-7}] - \phi_7 \underset{t}{\mathbb{E}}[X_{t+h-19}] +$ $\phi_8 \underset{t}{\mathbb{E}}[X_{t+h-8}] - \phi_8 \underset{t}{\mathbb{E}}[X_{t+h-20}] +$ $\phi_9 \underset{t}{\mathbb{E}}[X_{t+h-9}] - \phi_9 \underset{t}{\mathbb{E}}[X_{t+h-21}] +$ $\phi_{10} \underset{t}{\mathbb{E}}[X_{t+h-10}] - \phi_{10} \underset{t}{\mathbb{E}}[X_{t+h-22}] +$ $\phi_{11} \underset{t}{\mathbb{E}}[X_{t+h-11}] - \phi_{11} \underset{t}{\mathbb{E}}[X_{t+h-23}] +$ $\phi_{12} \underset{t}{\mathbb{E}}[X_{t+h-12}] - \phi_{12} \underset{t}{\mathbb{E}}[X_{t+h-24}] +$ $\phi_{13} \underset{t}{\mathbb{E}}[X_{t+h-13}] - \phi_{13} \underset{t}{\mathbb{E}}[X_{t+h-25}]$

El polinomio desarrollado y con las $\Theta_j$ distribuidas:

$- \Theta_1 \underset{t}{\mathbb{E}}[\varepsilon_{t+h-12}] -$ $\Theta_2 \underset{t}{\mathbb{E}}[\varepsilon_{t+h-24}] -$ $\Theta_3 \underset{t}{\mathbb{E}}[\varepsilon_{t+h-36}] -$ $\Theta_4 \underset{t}{\mathbb{E}}[\varepsilon_{t+h-48}]$

PRONÓSTICO ÓPTIMO¶

Se obtuvieron en Excel

INTERVALO DE CONFIANZA¶

CALCULO DE LAS $\psi$¶

ψ0 = -1
ψ1 = -0.14196630827330695
ψ2 = -0.0012833831644303538
ψ3 = -0.04711945564958303
ψ4 = 0.055666495988799594
ψ5 = 0.13168502721815675
ψ6 = 0.14390060444632985
ψ7 = 0.05528005326826947
ψ8 = 0.06597563860600197
ψ9 = -0.023742200999115167
ψ10 = -0.029931708277211113
ψ11 = -0.13198547391317694
ψ12 = 0.7710272345141591

DESVIACIÓN ESTANDAR DEL RUIDO BLANCO¶

0.6807672997433546

INTERVALOS DE PREDICCIÓN¶

Intervalo 1: (-1.393982459496975, 1.2746253554969749)
Intervalo 2: (-1.9878485554111043, 0.7075172474111042)
Intervalo 3: (-1.6506852053494947, 1.0446827733494948)
Intervalo 4: (-1.3467435497585691, 1.3515559017585692)
Intervalo 5: (-1.2722535341171932, 1.4301320121171934)
Intervalo 6: (-0.06293200205383043, 2.6622066420538304)
Intervalo 7: (0.7020341131440195, 3.45409653885598)
Intervalo 8: (-0.349748182450476, 2.406265238450476)
Intervalo 9: (-0.9982177354298769, 1.7634137014298767)
Intervalo 10: (-0.7469687343170736, 2.0153894083170734)
Intervalo 11: (-1.4538500316205047, 1.3096627116205048)
Intervalo 12: (-2.5035084583313623, 0.2823594183313627)
Pronóstico Límite Inferior Límite Superior Real Residuos
2008-12-31 731.469977 -443.310715 1906.250669 74 -657.469977
2009-01-31 220.383233 -966.176922 1406.943388 148 -72.383233
2009-02-28 517.237847 -669.323266 1703.798960 8 -509.237847
2009-03-31 786.132141 -401.719472 1973.983754 64 -722.132141
2009-04-30 853.515264 -336.135138 2043.165667 436 -417.515264
2009-05-31 1928.272318 728.605494 3127.939142 2627 698.727682
2009-06-30 2613.635065 1402.115792 3825.154339 972 -1641.635065
2009-07-31 1689.338322 476.079732 2902.596911 1474 -215.338322
2009-08-31 1120.869971 -94.861795 2336.601736 3161 2040.130029
2009-09-30 1342.400713 126.349036 2558.452391 1190 -152.400713
2009-10-31 720.539162 -496.020797 1937.099120 155 -565.539162
2009-11-30 -193.785721 -1420.186908 1032.615466 51 244.785721
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COMPARACIÓN¶

Pronóstico Óptimo Método Forecast Real Residuos Óptimo Residuos Forecast
2008-12-31 731.469977 540.502158 74 -657.469977 -466.502158
2009-01-31 220.383233 201.870616 148 -72.383233 -53.870616
2009-02-28 517.237847 218.349640 8 -509.237847 -210.349640
2009-03-31 786.132141 666.919279 64 -722.132141 -602.919279
2009-04-30 853.515264 824.911936 436 -417.515264 -388.911936
2009-05-31 1928.272318 1406.446553 2627 698.727682 1220.553447
2009-06-30 2613.635065 1256.178392 972 -1641.635065 -284.178392
2009-07-31 1689.338322 1262.636919 1474 -215.338322 211.363081
2009-08-31 1120.869971 1259.408611 3161 2040.130029 1901.591389
2009-09-30 1342.400713 736.647934 1190 -152.400713 453.352066
2009-10-31 720.539162 307.230864 155 -565.539162 -152.230864
2009-11-30 -193.785721 340.113522 51 244.785721 -289.113522
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ACTUALIZACIÓN DE PRONÓSTICO¶

CON PYTHON¶

Pronóstico Pronostico Actualizado Valores Reales Residuos Residuos Actualizados
2009-01-31 201.870616 135.678753 148 -53.870616 12.321247
2009-02-28 218.349640 217.762493 8 -210.349640 -209.762493
2009-03-31 666.919279 644.960938 64 -602.919279 -580.960938
2009-04-30 824.911936 850.856468 436 -388.911936 -414.856468
2009-05-31 1406.446553 1467.835729 2627 1220.553447 1159.164271
2009-06-30 1256.178392 1323.237743 972 -284.178392 -351.237743
2009-07-31 1262.636919 1288.347469 1474 211.363081 185.652531
2009-08-31 1259.408611 1290.129012 3161 1901.591389 1870.870988
2009-09-30 736.647934 725.578659 1190 453.352066 464.421341
2009-10-31 307.230864 293.280258 155 -152.230864 -138.280258
2009-11-30 340.113522 278.605347 51 -289.113522 -227.605347
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MANUAL¶

Pronóstico Óptimo Pronóstico Actualizado Valores Reales
2008-12-31 731.469977 NaN 74
2009-01-31 220.383233 74.000000 148
2009-02-28 517.237847 127.044647 8
2009-03-31 786.132141 516.394061 64
2009-04-30 853.515264 755.152514 436
2009-05-31 1928.272318 890.114314 2627
2009-06-30 2613.635065 2014.851270 972
2009-07-31 1689.338322 2708.245392 1474
2009-08-31 1120.869971 1725.683297 3161
2009-09-30 1342.400713 1164.246972 1190
2009-10-31 720.539162 1326.790929 155
2009-11-30 -193.785721 700.859962 51
No description has been provided for this image

COMPARACIÓN¶

Pronóstico Óptimo Pronóstico Actualizado Método Forecast Método Forecast Actualizado Valores Reales
2008-12-31 731.469977 NaN 540.502158 NaN 74
2009-01-31 220.383233 74.000000 201.870616 201.870616 148
2009-02-28 517.237847 127.044647 218.349640 218.349640 8
2009-03-31 786.132141 516.394061 666.919279 666.919279 64
2009-04-30 853.515264 755.152514 824.911936 824.911936 436
2009-05-31 1928.272318 890.114314 1406.446553 1406.446553 2627
2009-06-30 2613.635065 2014.851270 1256.178392 1256.178392 972
2009-07-31 1689.338322 2708.245392 1262.636919 1262.636919 1474
2009-08-31 1120.869971 1725.683297 1259.408611 1259.408611 3161
2009-09-30 1342.400713 1164.246972 736.647934 736.647934 1190
2009-10-31 720.539162 1326.790929 307.230864 307.230864 155
2009-11-30 -193.785721 700.859962 340.113522 340.113522 51
No description has been provided for this image